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.side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Appendix:Glossary_of_abstract_algebra" class="extiw" title="wiktionary:Appendix:Glossary of abstract algebra">Appendix:Glossary of abstract algebra</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <p><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a> is the subject area of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> that studies <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a>, such as <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>, <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, and <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebras</a>. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, often now called <i><a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a></i>. The distinction is rarely made in more recent writings. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_language">Basic language</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=1" title="Edit section: Basic language"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a> are defined primarily as <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> with <i>operations</i>. </p> <ul><li><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> <ul><li><a href="/wiki/Subobject" title="Subobject">Subobjects</a>: <a href="/wiki/Subgroup" title="Subgroup">subgroup</a>, <a href="/wiki/Subring" title="Subring">subring</a>, <a href="/wiki/Subalgebra" title="Subalgebra">subalgebra</a>, <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodule</a> etc.</li></ul></li> <li><a href="/wiki/Binary_operation" title="Binary operation">Binary operation</a> <ul><li><a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">Closure</a> of an operation</li> <li><a href="/wiki/Associative_property" title="Associative property">Associative property</a></li> <li><a href="/wiki/Distributive_property" title="Distributive property">Distributive property</a></li> <li><a href="/wiki/Commutative_property" title="Commutative property">Commutative property</a></li></ul></li> <li><a href="/wiki/Unary_operator" class="mw-redirect" title="Unary operator">Unary operator</a> <ul><li><a href="/wiki/Additive_inverse" title="Additive inverse">Additive inverse</a>, <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>, <a href="/wiki/Inverse_element" title="Inverse element">inverse element</a> <ul><li><a href="/wiki/Identity_element" title="Identity element">Identity element</a></li> <li><a href="/wiki/Cancellation_property" title="Cancellation property">Cancellation property</a></li></ul></li></ul></li> <li><a href="/wiki/Finitary_operation" class="mw-redirect" title="Finitary operation">Finitary operation</a> <ul><li><a href="/wiki/Arity" title="Arity">Arity</a></li></ul></li></ul> <p>Structure preserving maps called <i>homomorphisms</i> are vital in the study of algebraic objects. </p> <ul><li><a href="/wiki/Homomorphism" title="Homomorphism">Homomorphisms</a> <ul><li><a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">Kernels</a> and <a href="/wiki/Cokernel" title="Cokernel">cokernels</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">Image</a> and <a href="/wiki/Coimage" title="Coimage">coimage</a></li> <li><a href="/wiki/Epimorphism" title="Epimorphism">Epimorphisms</a> and <a href="/wiki/Monomorphism" title="Monomorphism">monomorphisms</a> <ul><li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphisms</a> <ul><li><a href="/wiki/Isomorphism_theorem" class="mw-redirect" title="Isomorphism theorem">Isomorphism theorems</a></li></ul></li></ul></li></ul></li></ul> <p>There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra. </p> <ul><li><a href="/wiki/Direct_sum" title="Direct sum">Direct sum</a> <ul><li><a href="/wiki/Direct_limit" title="Direct limit">Direct limit</a></li></ul></li> <li><a href="/wiki/Direct_product" title="Direct product">Direct product</a> <ul><li><a href="/wiki/Inverse_limit" title="Inverse limit">Inverse limit</a></li></ul></li> <li>Quotient objects: <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a>, <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a>, <a href="/wiki/Quotient_module" title="Quotient module">quotient module</a> etc.</li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> <p>Advanced concepts: </p> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a> <ul><li><a href="/wiki/Category_of_groups" title="Category of groups">Category of groups</a> <ul><li><a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">Category of abelian groups</a></li></ul></li> <li><a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a></li> <li><a href="/wiki/Category_of_modules" title="Category of modules">Category of modules</a> (over a fixed ring) <ul><li><a href="/wiki/Morita_equivalence" title="Morita equivalence">Morita equivalence</a>, <a href="/wiki/Morita_equivalence#Further_directions" title="Morita equivalence">Morita duality</a></li></ul></li> <li><a href="/wiki/Category_of_vector_spaces" class="mw-redirect" title="Category of vector spaces">Category of vector spaces</a></li></ul></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a> <ul><li><a href="/wiki/Filtration_(algebra)" class="mw-redirect" title="Filtration (algebra)">Filtration (algebra)</a></li> <li><a href="/wiki/Exact_sequence" title="Exact sequence">Exact sequence</a></li> <li><a href="/wiki/Functor" title="Functor">Functor</a></li></ul></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Semigroups_and_monoids">Semigroups and monoids</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=2" title="Edit section: Semigroups and monoids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> <ul><li><a href="/wiki/Subsemigroup" class="mw-redirect" title="Subsemigroup">Subsemigroup</a></li> <li><a href="/wiki/Free_semigroup" class="mw-redirect" title="Free semigroup">Free semigroup</a></li> <li><a href="/wiki/Green%27s_relations" title="Green's relations">Green's relations</a></li> <li><a href="/wiki/Inverse_semigroup" title="Inverse semigroup">Inverse semigroup</a> (or inversion semigroup, cf. <a rel="nofollow" class="external autonumber" href="http://www.encyclopediaofmath.org/index.php/Inversion_semi-group">[1]</a>)</li> <li><a href="/wiki/Krohn%E2%80%93Rhodes_theory" title="Krohn–Rhodes theory">Krohn–Rhodes theory</a></li> <li><a href="/wiki/Semigroup_algebra" class="mw-redirect" title="Semigroup algebra">Semigroup algebra</a></li> <li><a href="/wiki/Transformation_semigroup" title="Transformation semigroup">Transformation semigroup</a></li></ul></li> <li><a href="/wiki/Monoid" title="Monoid">Monoid</a> <ul><li><a href="/wiki/Aperiodic_monoid" class="mw-redirect" title="Aperiodic monoid">Aperiodic monoid</a></li> <li><a href="/wiki/Free_monoid" title="Free monoid">Free monoid</a></li> <li><a href="/wiki/Monoid_(category_theory)" title="Monoid (category theory)">Monoid (category theory)</a></li> <li><a href="/wiki/Monoid_factorisation" title="Monoid factorisation">Monoid factorisation</a></li> <li><a href="/wiki/Syntactic_monoid" title="Syntactic monoid">Syntactic monoid</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Group_theory">Group theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=3" title="Edit section: Group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></div> <dl><dt>Structure</dt></dl> <ul><li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group (mathematics)</a> <ul><li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem (group theory)</a></li> <li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a> <ul><li><a href="/wiki/Coset" title="Coset">Coset</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Characteristic_subgroup" title="Characteristic subgroup">Characteristic subgroup</a></li> <li><a href="/wiki/Centralizer_and_normalizer" title="Centralizer and normalizer">Centralizer and normalizer</a> subgroups</li> <li><a href="/wiki/Derived_group" class="mw-redirect" title="Derived group">Derived group</a></li> <li><a href="/wiki/Frattini_subgroup" title="Frattini subgroup">Frattini subgroup</a></li> <li><a href="/wiki/Fitting_subgroup" title="Fitting subgroup">Fitting subgroup</a></li></ul></li> <li><a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></li> <li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Local_analysis" title="Local analysis">Local analysis</a></li></ul></li></ul> <dl><dt>Constructions</dt></dl> <ul><li><a href="/wiki/Free_group" title="Free group">Free group</a> <ul><li><a href="/wiki/Presentation_of_a_group" title="Presentation of a group">Presentation of a group</a></li> <li><a href="/wiki/Word_problem_for_groups" title="Word problem for groups">Word problem for groups</a></li></ul></li> <li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a> <ul><li><a href="/wiki/Extension_problem" class="mw-redirect" title="Extension problem">Extension problem</a></li></ul></li> <li><a href="/wiki/Direct_sum" title="Direct sum">Direct sum</a>, <a href="/wiki/Direct_product" title="Direct product">direct product</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">Semidirect product</a> <ul><li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></li></ul> <dl><dt>Types</dt></dl> <ul><li><a href="/wiki/Simple_group" title="Simple group">Simple group</a></li> <li><a href="/wiki/Finite_group" title="Finite group">Finite group</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a> <ul><li><a href="/wiki/Torsion_subgroup" title="Torsion subgroup">Torsion subgroup</a></li> <li><a href="/wiki/Free_abelian_group" title="Free abelian group">Free abelian group</a></li> <li><a href="/wiki/Finitely_generated_abelian_group" title="Finitely generated abelian group">Finitely generated abelian group</a></li> <li><a href="/wiki/Rank_of_an_abelian_group" title="Rank of an abelian group">Rank of an abelian group</a></li></ul></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> <ul><li><a href="/wiki/Locally_cyclic_group" title="Locally cyclic group">Locally cyclic group</a></li></ul></li> <li><a href="/wiki/Solvable_group" title="Solvable group">Solvable group</a> <ul><li><a href="/wiki/Composition_series" title="Composition series">Composition series</a></li></ul></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">Nilpotent group</a></li> <li><a href="/wiki/Divisible_group" title="Divisible group">Divisible group</a></li> <li><a href="/wiki/Dedekind_group" title="Dedekind group">Dedekind group</a>, <a href="/wiki/Hamiltonian_group" class="mw-redirect" title="Hamiltonian group">Hamiltonian group</a></li></ul> <dl><dt>Examples</dt></dl> <ul><li><a href="/wiki/Examples_of_groups" title="Examples of groups">Examples of groups</a> <ul><li><a href="/wiki/Trivial_group" title="Trivial group">Trivial group</a></li> <li><a href="/wiki/Additive_group" title="Additive group">Additive group</a></li> <li><a href="/wiki/Permutation_group" title="Permutation group">Permutation group</a></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a></li> <li><a href="/wiki/P-group" title="P-group">p-group</a></li> <li><a href="/wiki/List_of_small_groups" title="List of small groups">List of small groups</a></li> <li><a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a></li> <li><a href="/wiki/Dicyclic_group" title="Dicyclic group">Dicyclic group</a></li> <li><a href="/wiki/Automorphism_group" title="Automorphism group">Automorphism group</a></li> <li><a href="/wiki/Point_group" title="Point group">Point group</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle group</a></li> <li><a href="/wiki/Linear_group" title="Linear group">Linear group</a></li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal group</a></li></ul></li></ul> <dl><dt>Applications</dt></dl> <ul><li><a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">Group action</a> <ul><li><a href="/wiki/Conjugacy_class" title="Conjugacy class">Conjugacy class</a></li> <li><a href="/wiki/Inner_automorphism" title="Inner automorphism">Inner automorphism</a></li> <li><a href="/wiki/Conjugate_closure" class="mw-redirect" title="Conjugate closure">Conjugate closure</a></li> <li><a href="/wiki/Stabilizer_subgroup" class="mw-redirect" title="Stabilizer subgroup">Stabilizer subgroup</a></li> <li><a href="/wiki/Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">Orbit (group theory)</a></li> <li><a href="/wiki/Orbit-stabilizer_theorem" class="mw-redirect" title="Orbit-stabilizer theorem">Orbit-stabilizer theorem</a></li> <li><a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a></li> <li><a href="/wiki/Burnside%27s_lemma" title="Burnside's lemma">Burnside's lemma</a></li></ul></li> <li><a href="/wiki/Burnside%27s_problem" class="mw-redirect" title="Burnside's problem">Burnside's problem</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop group</a></li> <li><a href="/wiki/Fundamental_group" title="Fundamental group">Fundamental group</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ring_theory">Ring theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=4" title="Edit section: Ring theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></div> <dl><dt>General</dt></dl> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring (mathematics)</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a>, <a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a>, <a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative ring</a></li> <li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra over a field</a> <ul><li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative algebra</a></li></ul></li> <li>Relatives to rings: <a href="/wiki/Semiring" title="Semiring">Semiring</a>, <a href="/wiki/Nearring" class="mw-redirect" title="Nearring">Nearring</a>, <a href="/wiki/Rig_(algebra)" class="mw-redirect" title="Rig (algebra)">Rig (algebra)</a></li></ul> <dl><dt>Structure</dt></dl> <ul><li><a href="/wiki/Subring" title="Subring">Subring</a>, <a href="/wiki/Subalgebra" title="Subalgebra">Subalgebra</a> <ul><li><a href="/wiki/Center_(algebra)" title="Center (algebra)">Center (algebra)</a></li></ul></li> <li><a href="/wiki/Ring_ideal" class="mw-redirect" title="Ring ideal">Ring ideal</a> <ul><li><a href="/wiki/Principal_ideal" title="Principal ideal">Principal ideal</a></li> <li><a href="/wiki/Ideal_quotient" title="Ideal quotient">Ideal quotient</a></li> <li><a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">Maximal ideal</a>, <a href="/wiki/Minimal_ideal" title="Minimal ideal">minimal ideal</a></li> <li><a href="/wiki/Primitive_ideal" title="Primitive ideal">Primitive ideal</a>, <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>, <a href="/wiki/Semiprime_ideal" class="mw-redirect" title="Semiprime ideal">semiprime ideal</a></li> <li><a href="/wiki/Radical_of_an_ideal" title="Radical of an ideal">Radical of an ideal</a></li></ul></li> <li><a href="/wiki/Jacobson_radical" title="Jacobson radical">Jacobson radical</a></li> <li><a href="/wiki/Socle_of_a_ring" class="mw-redirect" title="Socle of a ring">Socle of a ring</a></li> <li><a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit (ring theory)</a>, <a href="/wiki/Idempotent_element_(ring_theory)" class="mw-redirect" title="Idempotent element (ring theory)">Idempotent</a>, <a href="/wiki/Nilpotent" title="Nilpotent">Nilpotent</a>, <a href="/wiki/Zero_divisor" title="Zero divisor">Zero divisor</a></li> <li><a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">Characteristic (algebra)</a></li> <li><a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphism</a>, <a href="/wiki/Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">Algebra homomorphism</a> <ul><li><a href="/wiki/Ring_epimorphism" class="mw-redirect" title="Ring epimorphism">Ring epimorphism</a></li> <li><a href="/wiki/Ring_monomorphism" class="mw-redirect" title="Ring monomorphism">Ring monomorphism</a></li> <li><a href="/wiki/Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">Ring isomorphism</a> <ul><li><a href="/wiki/Skolem%E2%80%93Noether_theorem" title="Skolem–Noether theorem">Skolem–Noether theorem</a></li></ul></li></ul></li> <li><a href="/wiki/Graded_algebra" class="mw-redirect" title="Graded algebra">Graded algebra</a></li> <li><a href="/wiki/Morita_equivalence" title="Morita equivalence">Morita equivalence</a> <ul><li><a href="/wiki/Brauer_group" title="Brauer group">Brauer group</a></li></ul></li></ul> <dl><dt>Constructions</dt></dl> <ul><li><a href="/wiki/Direct_sum_of_rings" class="mw-redirect" title="Direct sum of rings">Direct sum of rings</a>, <a href="/wiki/Product_of_rings" title="Product of rings">Product of rings</a></li> <li><a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a></li> <li><a href="/wiki/Matrix_ring" title="Matrix ring">Matrix ring</a></li> <li><a href="/wiki/Endomorphism_ring" title="Endomorphism ring">Endomorphism ring</a></li> <li><a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li> <li><a href="/wiki/Monoid_ring" title="Monoid ring">Monoid ring</a>, <a href="/wiki/Group_ring" title="Group ring">Group ring</a></li> <li><a href="/wiki/Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">Localization of a ring</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a> <ul><li><a href="/wiki/Symmetric_algebra" title="Symmetric algebra">Symmetric algebra</a>, <a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a>, <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li> <li><a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a></li> <li><a href="/wiki/Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">Completion (ring theory)</a></li></ul> <dl><dt>Types</dt></dl> <ul><li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field (mathematics)</a>, <a href="/wiki/Division_ring" title="Division ring">Division ring</a>, <a href="/wiki/Division_algebra" title="Division algebra">division algebra</a></li> <li><a href="/wiki/Simple_ring" title="Simple ring">Simple ring</a>, <a href="/wiki/Central_simple_algebra" title="Central simple algebra">Central simple algebra</a>, <a href="/wiki/Semisimple_ring" class="mw-redirect" title="Semisimple ring">Semisimple ring</a>, <a href="/wiki/Semisimple_algebra" title="Semisimple algebra">Semisimple algebra</a></li> <li><a href="/wiki/Primitive_ring" title="Primitive ring">Primitive ring</a>, <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></li> <li><a href="/wiki/Prime_ring" title="Prime ring">Prime ring</a>, <a href="/wiki/Semiprime_ring" title="Semiprime ring">Semiprime ring</a>, <a href="/wiki/Reduced_ring" title="Reduced ring">Reduced ring</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a>, <a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain (ring theory)</a> <ul><li><a href="/wiki/Field_of_fractions" title="Field of fractions">Field of fractions</a>, <a href="/wiki/Integral_closure" class="mw-redirect" title="Integral closure">Integral closure</a></li> <li><a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a>, <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a>, <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a>, <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domain</a>, <a href="/wiki/Pr%C3%BCfer_domain" title="Prüfer domain">Prüfer domain</a></li></ul></li> <li><a href="/wiki/Von_Neumann_regular_ring" title="Von Neumann regular ring">Von Neumann regular ring</a></li> <li><a href="/wiki/Quasi-Frobenius_ring" title="Quasi-Frobenius ring">Quasi-Frobenius ring</a></li> <li><a href="/wiki/Hereditary_ring" title="Hereditary ring">Hereditary ring</a>, <a href="/wiki/Semihereditary_ring" class="mw-redirect" title="Semihereditary ring">Semihereditary ring</a></li> <li><a href="/wiki/Local_ring" title="Local ring">Local ring</a>, <a href="/wiki/Semi-local_ring" title="Semi-local ring">Semi-local ring</a></li> <li><a href="/wiki/Discrete_valuation_ring" title="Discrete valuation ring">Discrete valuation ring</a></li> <li><a href="/wiki/Regular_local_ring" title="Regular local ring">Regular local ring</a></li> <li><a href="/wiki/Cohen%E2%80%93Macaulay_ring" title="Cohen–Macaulay ring">Cohen–Macaulay ring</a></li> <li><a href="/wiki/Gorenstein_ring" title="Gorenstein ring">Gorenstein ring</a></li> <li><a href="/wiki/Artinian_ring" title="Artinian ring">Artinian ring</a>, <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian ring</a></li> <li><a href="/wiki/Perfect_ring" title="Perfect ring">Perfect ring</a>, <a href="/wiki/Semiperfect_ring" class="mw-redirect" title="Semiperfect ring">semiperfect ring</a></li> <li><a href="/wiki/Baer_ring" title="Baer ring">Baer ring</a>, <a href="/wiki/Rickart_ring" class="mw-redirect" title="Rickart ring">Rickart ring</a></li> <li><a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a>, <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> <ul><li><a href="/wiki/Ideal_(Lie_algebra)" class="mw-redirect" title="Ideal (Lie algebra)">Ideal (Lie algebra)</a></li></ul></li> <li><a href="/wiki/Jordan_algebra" title="Jordan algebra">Jordan algebra</a></li> <li><a href="/wiki/Differential_algebra" title="Differential algebra">Differential algebra</a></li> <li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li></ul> <dl><dt>Examples</dt></dl> <ul><li><a href="/wiki/Rational_number" title="Rational number">Rational number</a>, <a href="/wiki/Real_number" title="Real number">Real number</a>, <a href="/wiki/Complex_number" title="Complex number">Complex number</a>, <a href="/wiki/Quaternions" class="mw-redirect" title="Quaternions">Quaternions</a>, <a href="/wiki/Octonions" class="mw-redirect" title="Octonions">Octonions</a> <ul><li><a href="/wiki/Hurwitz_quaternion" title="Hurwitz quaternion">Hurwitz quaternion</a></li></ul></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li></ul> <dl><dt>Theorems and applications</dt></dl> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> <ul><li><a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a></li></ul></li> <li><a href="/wiki/Hilbert%27s_basis_theorem" title="Hilbert's basis theorem">Hilbert's basis theorem</a></li> <li><a href="/wiki/Hopkins%E2%80%93Levitzki_theorem" title="Hopkins–Levitzki theorem">Hopkins–Levitzki theorem</a></li> <li><a href="/wiki/Krull%27s_principal_ideal_theorem" title="Krull's principal ideal theorem">Krull's principal ideal theorem</a></li> <li><a href="/wiki/Levitzky%27s_theorem" title="Levitzky's theorem">Levitzky's theorem</a></li> <li><a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> <ul><li><a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a></li></ul></li> <li><a href="/wiki/Artin-Wedderburn_theorem" class="mw-redirect" title="Artin-Wedderburn theorem">Artin-Wedderburn theorem</a></li> <li><a href="/wiki/Jacobson_density_theorem" title="Jacobson density theorem">Jacobson density theorem</a></li> <li><a href="/wiki/Wedderburn%27s_little_theorem" title="Wedderburn's little theorem">Wedderburn's little theorem</a></li> <li><a href="/wiki/Lasker%E2%80%93Noether_theorem" class="mw-redirect" title="Lasker–Noether theorem">Lasker–Noether theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Field_theory">Field theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=5" title="Edit section: Field theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">Field theory (mathematics)</a></div> <dl><dt>Basic concepts</dt></dl> <ul><li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field (mathematics)</a></li> <li><a href="/wiki/Subfield_(mathematics)" class="mw-redirect" title="Subfield (mathematics)">Subfield (mathematics)</a> <ul><li><a href="/wiki/Multiplicative_group" title="Multiplicative group">Multiplicative group</a> <ul><li><a href="/wiki/Primitive_element_(field_theory)" class="mw-redirect" title="Primitive element (field theory)">Primitive element (field theory)</a></li></ul></li></ul></li> <li><a href="/wiki/Field_extension" title="Field extension">Field extension</a> <ul><li><a href="/wiki/Algebraic_extension" title="Algebraic extension">Algebraic extension</a> <ul><li><a href="/wiki/Splitting_field" title="Splitting field">Splitting field</a></li> <li><a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">Algebraically closed field</a> <ul><li><a href="/wiki/Algebraic_element" title="Algebraic element">Algebraic element</a></li> <li><a href="/wiki/Algebraic_closure" title="Algebraic closure">Algebraic closure</a></li></ul></li></ul></li> <li><a href="/wiki/Separable_extension" title="Separable extension">Separable extension</a> <ul><li><a href="/wiki/Separable_polynomial" title="Separable polynomial">Separable polynomial</a></li></ul></li> <li><a href="/wiki/Normal_extension" title="Normal extension">Normal extension</a></li> <li><a href="/wiki/Galois_extension" title="Galois extension">Galois extension</a></li> <li><a href="/wiki/Abelian_extension" title="Abelian extension">Abelian extension</a></li> <li><a href="/wiki/Transcendence_degree" class="mw-redirect" title="Transcendence degree">Transcendence degree</a></li></ul></li> <li><a href="/wiki/Field_norm" title="Field norm">Field norm</a></li> <li><a href="/wiki/Field_trace" title="Field trace">Field trace</a></li> <li><a href="/wiki/Conjugate_element_(field_theory)" title="Conjugate element (field theory)">Conjugate element (field theory)</a></li> <li><a href="/wiki/Tensor_product_of_fields" title="Tensor product of fields">Tensor product of fields</a></li></ul> <dl><dt>Types</dt></dl> <ul><li><a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></li> <li><a href="/wiki/Global_field" title="Global field">Global field</a></li> <li><a href="/wiki/Local_field" title="Local field">Local field</a></li> <li><a href="/wiki/Finite_field" title="Finite field">Finite field</a></li> <li><a href="/wiki/Symmetric_function" title="Symmetric function">Symmetric function</a></li> <li><a href="/wiki/Formally_real_field" title="Formally real field">Formally real field</a></li> <li><a href="/wiki/Real_closed_field" title="Real closed field">Real closed field</a></li></ul> <dl><dt>Applications</dt></dl> <ul><li><a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> <ul><li><a href="/wiki/Galois_group" title="Galois group">Galois group</a></li> <li><a href="/wiki/Inverse_Galois_problem" title="Inverse Galois problem">Inverse Galois problem</a></li></ul></li> <li><a href="/wiki/Kummer_theory" title="Kummer theory">Kummer theory</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Module_theory">Module theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=6" title="Edit section: Module theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module (mathematics)</a></div> <dl><dt>General</dt></dl> <ul><li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module (mathematics)</a></li> <li><a href="/wiki/Bimodule" title="Bimodule">Bimodule</a></li> <li><a href="/wiki/Annihilator_(ring_theory)" title="Annihilator (ring theory)">Annihilator (ring theory)</a></li></ul> <dl><dt>Structure</dt></dl> <ul><li><a href="/wiki/Submodule" class="mw-redirect" title="Submodule">Submodule</a> <ul><li><a href="/wiki/Pure_submodule" title="Pure submodule">Pure submodule</a></li></ul></li> <li><a href="/wiki/Module_homomorphism" title="Module homomorphism">Module homomorphism</a> <ul><li><a href="/wiki/Essential_submodule" class="mw-redirect" title="Essential submodule">Essential submodule</a></li> <li><a href="/wiki/Superfluous_submodule" class="mw-redirect" title="Superfluous submodule">Superfluous submodule</a></li> <li><a href="/wiki/Singular_submodule" title="Singular submodule">Singular submodule</a></li></ul></li> <li><a href="/wiki/Socle_of_a_module" class="mw-redirect" title="Socle of a module">Socle of a module</a></li> <li><a href="/wiki/Radical_of_a_module" title="Radical of a module">Radical of a module</a></li></ul> <dl><dt>Constructions</dt></dl> <ul><li><a href="/wiki/Free_module" title="Free module">Free module</a></li> <li><a href="/wiki/Quotient_module" title="Quotient module">Quotient module</a></li> <li><a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">Direct sum</a>, <a href="/wiki/Direct_product_of_modules" class="mw-redirect" title="Direct product of modules">Direct product of modules</a></li> <li><a href="/wiki/Direct_limit" title="Direct limit">Direct limit</a>, <a href="/wiki/Inverse_limit" title="Inverse limit">Inverse limit</a></li> <li><a href="/wiki/Localization_of_a_module" class="mw-redirect" title="Localization of a module">Localization of a module</a></li> <li><a href="/wiki/Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">Completion (ring theory)</a></li></ul> <dl><dt>Types</dt></dl> <ul><li><a href="/wiki/Simple_module" title="Simple module">Simple module</a>, <a href="/wiki/Semisimple_module" title="Semisimple module">Semisimple module</a></li> <li><a href="/wiki/Indecomposable_module" title="Indecomposable module">Indecomposable module</a></li> <li><a href="/wiki/Artinian_module" title="Artinian module">Artinian module</a>, <a href="/wiki/Noetherian_module" title="Noetherian module">Noetherian module</a></li> <li>Homological types: <ul><li><a href="/wiki/Projective_module" title="Projective module">Projective module</a> <ul><li><a href="/wiki/Projective_cover" title="Projective cover">Projective cover</a></li> <li><a href="/wiki/Swan%27s_theorem" class="mw-redirect" title="Swan's theorem">Swan's theorem</a></li> <li><a href="/wiki/Quillen%E2%80%93Suslin_theorem" title="Quillen–Suslin theorem">Quillen–Suslin theorem</a></li></ul></li> <li><a href="/wiki/Injective_module" title="Injective module">Injective module</a> <ul><li><a href="/wiki/Injective_hull" title="Injective hull">Injective hull</a></li></ul></li> <li><a href="/wiki/Flat_module" title="Flat module">Flat module</a> <ul><li><a href="/wiki/Flat_cover" title="Flat cover">Flat cover</a></li></ul></li></ul></li> <li><a href="/wiki/Coherent_module" class="mw-redirect" title="Coherent module">Coherent module</a></li> <li><a href="/wiki/Finitely-generated_module" class="mw-redirect" title="Finitely-generated module">Finitely-generated module</a></li> <li><a href="/wiki/Finitely-presented_module" class="mw-redirect" title="Finitely-presented module">Finitely-presented module</a></li> <li><a href="/wiki/Finitely_related_module" class="mw-redirect" title="Finitely related module">Finitely related module</a></li> <li><a href="/wiki/Algebraically_compact_module" title="Algebraically compact module">Algebraically compact module</a></li> <li><a href="/wiki/Reflexive_module" class="mw-redirect" title="Reflexive module">Reflexive module</a></li></ul> <dl><dt>Concepts and theorems</dt></dl> <ul><li><a href="/wiki/Composition_series" title="Composition series">Composition series</a> <ul><li><a href="/wiki/Length_of_a_module" title="Length of a module">Length of a module</a></li></ul></li> <li><a href="/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain" title="Structure theorem for finitely generated modules over a principal ideal domain">Structure theorem for finitely generated modules over a principal ideal domain</a></li> <li><a href="/wiki/Homological_dimension" title="Homological dimension">Homological dimension</a> <ul><li><a href="/wiki/Projective_dimension" class="mw-redirect" title="Projective dimension">Projective dimension</a></li> <li><a href="/wiki/Injective_dimension" class="mw-redirect" title="Injective dimension">Injective dimension</a></li> <li><a href="/wiki/Flat_dimension" class="mw-redirect" title="Flat dimension">Flat dimension</a></li> <li><a href="/wiki/Global_dimension" title="Global dimension">Global dimension</a></li> <li><a href="/wiki/Weak_global_dimension" class="mw-redirect" title="Weak global dimension">Weak global dimension</a></li> <li><a href="/wiki/Cohomological_dimension" title="Cohomological dimension">Cohomological dimension</a></li></ul></li> <li><a href="/wiki/Krull_dimension" title="Krull dimension">Krull dimension</a></li> <li><a href="/wiki/Regular_sequence_(algebra)" class="mw-redirect" title="Regular sequence (algebra)">Regular sequence (algebra)</a>, <a href="/wiki/Depth_(algebra)" class="mw-redirect" title="Depth (algebra)">depth (algebra)</a></li> <li><a href="/wiki/Fitting_lemma" title="Fitting lemma">Fitting lemma</a></li> <li><a href="/wiki/Schur%27s_lemma" title="Schur's lemma">Schur's lemma</a></li> <li><a href="/wiki/Nakayama%27s_lemma" title="Nakayama's lemma">Nakayama's lemma</a></li> <li><a href="/wiki/Krull%E2%80%93Schmidt_theorem" title="Krull–Schmidt theorem">Krull–Schmidt theorem</a></li> <li><a href="/wiki/Steinitz_exchange_lemma" title="Steinitz exchange lemma">Steinitz exchange lemma</a></li> <li><a href="/wiki/Jordan%E2%80%93H%C3%B6lder_theorem" class="mw-redirect" title="Jordan–Hölder theorem">Jordan–Hölder theorem</a></li> <li><a href="/wiki/Artin%E2%80%93Rees_lemma" title="Artin–Rees lemma">Artin–Rees lemma</a></li> <li><a href="/wiki/Schanuel%27s_lemma" title="Schanuel's lemma">Schanuel's lemma</a></li> <li><a href="/wiki/Morita_equivalence" title="Morita equivalence">Morita equivalence</a> <ul><li><a href="/wiki/Progenerator" class="mw-redirect" title="Progenerator">Progenerator</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Representation_theory">Representation theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=7" title="Edit section: Representation theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_representation_theory_topics" title="List of representation theory topics">List of representation theory topics</a></div> <p>Representation theory </p> <ul><li><a href="/wiki/Algebra_representation" title="Algebra representation">Algebra representation</a></li> <li><a href="/wiki/Group_representation" title="Group representation">Group representation</a></li> <li><a href="/wiki/Lie_algebra_representation" title="Lie algebra representation">Lie algebra representation</a></li> <li><a href="/wiki/Maschke%27s_theorem" title="Maschke's theorem">Maschke's theorem</a></li> <li><a href="/wiki/Schur%27s_lemma" title="Schur's lemma">Schur's lemma</a></li> <li><a href="/wiki/Equivariant_map" title="Equivariant map">Equivariant map</a></li> <li><a href="/wiki/Frobenius_reciprocity" title="Frobenius reciprocity">Frobenius reciprocity</a> <ul><li><a href="/wiki/Induced_representation" title="Induced representation">Induced representation</a></li> <li><a href="/wiki/Restricted_representation" title="Restricted representation">Restricted representation</a></li></ul></li> <li><a href="/wiki/Affine_representation" title="Affine representation">Affine representation</a></li> <li><a href="/wiki/Projective_representation" title="Projective representation">Projective representation</a></li> <li><a href="/wiki/Modular_representation_theory" title="Modular representation theory">Modular representation theory</a></li> <li><a href="/wiki/Quiver_(mathematics)" title="Quiver (mathematics)">Quiver (mathematics)</a></li> <li><a href="/wiki/Representation_theory_of_Hopf_algebras" title="Representation theory of Hopf algebras">Representation theory of Hopf algebras</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Non-associative_systems">Non-associative systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=8" title="Edit section: Non-associative systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>General</dt></dl> <ul><li><a href="/wiki/Associative_property" title="Associative property">Associative property</a>, <a href="/wiki/Associator" title="Associator">Associator</a></li> <li><a href="/wiki/Heap_(mathematics)" title="Heap (mathematics)">Heap (mathematics)</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma (algebra)</a> <ul><li><a href="/wiki/Loop_(algebra)" class="mw-redirect" title="Loop (algebra)">Loop (algebra)</a>, <a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup</a></li></ul></li> <li><a href="/wiki/Nonassociative_ring" class="mw-redirect" title="Nonassociative ring">Nonassociative ring</a>, <a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative algebra</a> <ul><li><a href="/wiki/Universal_enveloping_algebra" title="Universal enveloping algebra">Universal enveloping algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> (<i>see also <a href="/wiki/List_of_Lie_group_topics" class="mw-redirect" title="List of Lie group topics">list of Lie group topics</a></i> and <a href="/wiki/List_of_representation_theory_topics" title="List of representation theory topics">list of representation theory topics</a>)</li> <li><a href="/wiki/Jordan_algebra" title="Jordan algebra">Jordan algebra</a></li> <li><a href="/wiki/Alternative_algebra" title="Alternative algebra">Alternative algebra</a></li> <li><a href="/wiki/Power_associativity" title="Power associativity">Power associativity</a></li> <li><a href="/wiki/Flexible_algebra" title="Flexible algebra">Flexible algebra</a></li></ul></li></ul> <dl><dt>Examples</dt></dl> <ul><li><a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a> <ul><li><a href="/wiki/Octonion" title="Octonion">Octonions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a></li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a></li></ul></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalities">Generalities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=9" title="Edit section: Generalities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal algebra</a> <ul><li><a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">Variety (universal algebra)</a></li> <li><a href="/wiki/Congruence_relation" title="Congruence relation">Congruence relation</a></li> <li><a href="/wiki/Free_object" title="Free object">Free object</a></li> <li><a href="/w/index.php?title=Generating_set_(universal_algebra)&action=edit&redlink=1" class="new" title="Generating set (universal algebra) (page does not exist)">Generating set (universal algebra)</a></li> <li><a href="/wiki/Clone_(algebra)" title="Clone (algebra)">Clone (algebra)</a></li></ul></li> <li><a href="/wiki/Kernel_of_a_function" class="mw-redirect" title="Kernel of a function">Kernel of a function</a> <ul><li><a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">Kernel (algebra)</a></li> <li><a href="/wiki/Isomorphism_class" class="mw-redirect" title="Isomorphism class">Isomorphism class</a></li> <li><a href="/wiki/Isomorphism_theorem" class="mw-redirect" title="Isomorphism theorem">Isomorphism theorem</a></li> <li><a href="/wiki/Fundamental_theorem_on_homomorphisms" title="Fundamental theorem on homomorphisms">Fundamental theorem on homomorphisms</a></li></ul></li> <li><a href="/wiki/Universal_property" title="Universal property">Universal property</a></li> <li><a href="/wiki/Filtration_(mathematics)" title="Filtration (mathematics)">Filtration (mathematics)</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a> <ul><li><a href="/wiki/Monoidal_category" title="Monoidal category">Monoidal category</a></li> <li><a href="/wiki/Groupoid" title="Groupoid">Groupoid</a></li> <li><a href="/wiki/Group_object" title="Group object">Group object</a></li> <li><a href="/wiki/Coalgebra" title="Coalgebra">Coalgebra</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li> <li><a href="/w/index.php?title=Magma_object&action=edit&redlink=1" class="new" title="Magma object (page does not exist)">Magma object</a></li></ul></li> <li><a href="/wiki/Torsion_(algebra)" title="Torsion (algebra)">Torsion (algebra)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Computer_algebra">Computer algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=10" title="Edit section: Computer algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Computer_algebra_system" title="Computer algebra system">Computer algebra system</a></div> <ul><li><a href="/wiki/Symbolic_mathematics" class="mw-redirect" title="Symbolic mathematics">Symbolic mathematics</a> <ul><li><a href="/wiki/Finite_field_arithmetic" title="Finite field arithmetic">Finite field arithmetic</a></li> <li><a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a></li> <li><a href="/wiki/Buchberger%27s_algorithm" title="Buchberger's algorithm">Buchberger's algorithm</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_abstract_algebra_topics&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_commutative_algebra_topics" title="List of commutative algebra topics">List of commutative algebra topics</a></li> <li><a href="/wiki/List_of_homological_algebra_topics" title="List of homological algebra topics">List of homological algebra topics</a></li> <li><a href="/wiki/List_of_linear_algebra_topics" class="mw-redirect" title="List of linear algebra topics">List of linear algebra topics</a></li> <li><a href="/wiki/List_of_algebraic_structures" class="mw-redirect" title="List of algebraic structures">List of algebraic structures</a></li> <li><a href="/wiki/Glossary_of_field_theory" title="Glossary of field theory">Glossary of field theory</a></li> <li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/Glossary_of_ring_theory" title="Glossary of ring theory">Glossary of ring theory</a></li> <li><a href="/wiki/Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary 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<li><a href="/wiki/History_of_algebra" title="History of algebra">History</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Areas</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> <ul><li><a href="/wiki/Algebraic_variety" title="Algebraic variety">Algebraic variety</a></li> <li><a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">Scheme</a></li></ul></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary algebra</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li> <li><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_expression" title="Algebraic expression">Algebraic expression</a></li> <li><a href="/wiki/Equation" title="Equation">Equation</a> (<a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a>, <a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a>)</li> <li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a> (<a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">Polynomial function</a>)</li> <li><a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">Inequality</a> (<a href="/wiki/Linear_inequality" title="Linear inequality">Linear inequality</a>)</li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> (<a href="/wiki/Addition" title="Addition">Addition</a>, <a href="/wiki/Multiplication" title="Multiplication">Multiplication</a>)</li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a>)</li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">Variable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic structures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> (<a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">theory</a>)</li> <li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a> (<a href="/wiki/Group_theory" title="Group theory">theory</a>)</li> <li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a> (<a href="/wiki/Commutative_algebra" title="Commutative algebra">theory</a>)</li> <li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a> (<a href="/wiki/Ring_theory" title="Ring theory">theory</a>)</li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a> (<a href="/wiki/Coordinate_vector" title="Coordinate vector">Vector</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Linear and <br />multilinear algebra</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a> (<a href="/wiki/Dot_product" title="Dot product">Dot product</a>)</li> <li><a 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constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Free_object" title="Free object">Free object</a> (<a href="/wiki/Free_group" title="Free group">Free group</a>, ...)</li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a> (<a href="/wiki/Multivector" title="Multivector">Multivector</a>)</li> <li><a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a> (<a href="/wiki/Polynomial" title="Polynomial">Polynomial</a>)</li> <li><a href="/wiki/Quotient_object" class="mw-redirect" title="Quotient object">Quotient object</a> (<a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a>, ...)</li> <li><a href="/wiki/Symmetric_algebra" 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