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id="for_a_ring">For a ring</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">S\subset R</annotation></semantics></math> is a left <a class="existingWikiWord" href="/nlab/show/Ore+set">Ore set</a> in a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (or a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, then we call the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(j,S^{-1}R)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>R</mi><mo>→</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi></mrow><annotation encoding="application/x-tex">j:R\to S^{-1}R</annotation></semantics></math> is a morphism of monoids (rings) the <strong>left Ore localization</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> if it is the universal object in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = C(R,S)</annotation></semantics></math> whose objects are the pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,Y)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : R \rightarrow Y</annotation></semantics></math> is a morphism of rings from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> into a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> such that the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(S)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> consists of units (=multiplicatively invertible elements), and the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>f</mi><mo>′</mo><mo>,</mo><mi>Y</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha : (f,Y) \rightarrow (f',Y')</annotation></semantics></math> are maps of rings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>Y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha : Y \rightarrow Y'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha \circ f = f'</annotation></semantics></math>.</p> <p>The definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> makes sense even if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">S\subset R</annotation></semantics></math> is not left Ore; the universal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> may then exist when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is not left Ore, for example this is the case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is right Ore, while not left Ore. In fact, the universal object is a left Ore localization (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is left Ore) iff it lies in the full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>l</mi></msup></mrow><annotation encoding="application/x-tex">C^l</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> whose objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,Y)</annotation></semantics></math> satisfy two additional conditions:</p> <p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>S</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>s</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">f(S)^{-1}f(R) = \{(f(s))^{-1}f(r)\,|\, s \in S, r\in R\}</annotation></semantics></math> is a subring in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>,</p> <p>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mspace width="thinmathspace"></mspace><mi>f</mi><mo>=</mo><msub><mi>I</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">ker\,f = I_S</annotation></semantics></math>.</p> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(j,S^{-1}R)</annotation></semantics></math> is universal in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>l</mi></msup></mrow><annotation encoding="application/x-tex">C^l</annotation></semantics></math>, and that characterizes it, but the universality in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, although not characteristic, appears to be more useful in practice.</p> <p>For every left Ore set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">S\subset R</annotation></semantics></math> in a monoid or ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, the left Ore localization exists and it can be defined as follows. As a set, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi><mo>:</mo><mo>=</mo><mi>S</mi><mo>×</mo><mi>R</mi><mo stretchy="false">/</mo><mo>∼</mo></mrow><annotation encoding="application/x-tex">S^{-1}R := S\times R/\sim</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> is the following relation of equivalence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>∼</mo><mo stretchy="false">(</mo><mi>s</mi><mo>′</mo><mo>,</mo><mi>r</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>⇔</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo>∃</mo><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>S</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>∃</mo><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><mi>s</mi><mo>′</mo><mo>=</mo><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><mi>s</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>and</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><mi>r</mi><mo>′</mo><mo>=</mo><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><mi>r</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (s,r) \sim (s',r') \,\,\Leftrightarrow\,\, (\exists \tilde s\in S \,\,\exists \tilde{r}\in R) \,\, (\tilde{s}s' = \tilde{r}s\,\,and\,\,\tilde{s}r' = \tilde{r}r). </annotation></semantics></math></div> <p>A class of equivalence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s,r)</annotation></semantics></math> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>s</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>r</mi></mrow><annotation encoding="application/x-tex">s^{-1}r</annotation></semantics></math> and called a left fraction. The multiplication is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>s</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>r</mi> <mn>1</mn></msub><mo>⋅</mo><msubsup><mi>s</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>r</mi> <mn>2</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><msub><mi>s</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_1^{-1}r_1\cdot s_2^{-1}r_2 = (\tilde{s}s_1)^{-1} (\tilde{r}r_2)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>R</mi><mo>,</mo><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\tilde{r} \in R, \tilde{s} \in S</annotation></semantics></math> satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><msub><mi>s</mi> <mn>2</mn></msub><mo>=</mo><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><msub><mi>r</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tilde{r}s_2 = \tilde{s}r_1</annotation></semantics></math> (one should think of this, though it is not yet formally justified at this point, as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>s</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><msub><mi>r</mi> <mn>1</mn></msub><msubsup><mi>s</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">s^{-1}\tilde{r} = r_1 s_2^{-1}</annotation></semantics></math>, what enables to put inverses one next to another and then the multiplication rule is obvious). If the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a ring then we can extend the addition to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi></mrow><annotation encoding="application/x-tex">S^{-1}R</annotation></semantics></math> too. Suppose we are given two fractions with representatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>r</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s_1,r_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s_2,r_2)</annotation></semantics></math>. Then by the left Ore condition we find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\tilde{s} \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\tilde{r}\in R</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><msub><mi>s</mi> <mn>1</mn></msub><mo>=</mo><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><msub><mi>s</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\tilde{s} s_1 = \tilde{r} s_2</annotation></semantics></math>. The sum is then defined</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>s</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>r</mi> <mn>1</mn></msub><mo>+</mo><msubsup><mi>s</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>r</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>:</mo><mo>=</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><msub><mi>s</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><msub><mi>r</mi> <mn>1</mn></msub><mo>+</mo><mover><mi>r</mi><mo stretchy="false">˜</mo></mover><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> s_1^{-1} r_1 + s_2^{-1} r_2 \,:=\, (\tilde{s}s_1)^{-1} (\tilde{s}r_1 + \tilde{r}r_2) </annotation></semantics></math></div> <p>It is a long and at points tricky to work out all the details of this definition. One has to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> is indeed relation of equivalence, that the operations are well defined, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi></mrow><annotation encoding="application/x-tex">S^{-1}R</annotation></semantics></math> is indeed a ring. Even the commutativity of the addition needs work (there is an alternative definition of addition in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>s</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{s}</annotation></semantics></math> above is not required to be in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> but the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>s</mi><mo stretchy="false">˜</mo></mover><msub><mi>r</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tilde{s}r_1</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>; this approach is manifestly commutative but it has some other drawbacks). At the end, one shows that the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>=</mo><msub><mi>j</mi> <mi>S</mi></msub><mo>:</mo><mi>R</mi><mo>→</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi></mrow><annotation encoding="application/x-tex">j = j_S : R \rightarrow S^{-1}R</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>r</mi></mrow><annotation encoding="application/x-tex">i(r) = 1^{-1}r</annotation></semantics></math> is a homomorphism of rings, which is 1-1 iff the 2-sided ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>S</mi></msub><mo>=</mo><mo stretchy="false">{</mo><mi>n</mi><mo>∈</mo><mi>R</mi><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mo>∃</mo><mi>s</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>sn</mi><mo>=</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I_S = \{ n \in R \,|\,\exists s \in S,\, sn = 0\}</annotation></semantics></math> is zero.</p> <h3 id="for_the_category_of_modules">For the category of modules</h3> <p>One defines a localization functor which is the extension of scalars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Q</mi> <mi>S</mi> <mo>*</mo></msubsup><mo>=</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi><msub><mo>⊗</mo> <mi>R</mi></msub><mo>−</mo><mo>:</mo><mi>R</mi><mo>−</mo><mi>mod</mi><mo>→</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi><mo>−</mo><mi>mod</mi></mrow><annotation encoding="application/x-tex">Q^*_S = S^{-1}R\otimes_R - : R-mod\to S^{-1}R-mod</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>↦</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi><msub><mo>⊗</mo> <mi>R</mi></msub><mi>M</mi></mrow><annotation encoding="application/x-tex">M\mapsto S^{-1}R\otimes_R M</annotation></semantics></math>. The localization functor is exact (“flat”), has a fully faithful right adjoint, namely the restriction of scalars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mrow><mi>S</mi><mo>*</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{S*}</annotation></semantics></math> and the latter has its own right adjoint (the localization functor is affine). In particular, it realizes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi><mo>−</mo><mi>mod</mi></mrow><annotation encoding="application/x-tex">S^{-1}R-mod</annotation></semantics></math> as a reflective subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>−</mo><mi>mod</mi></mrow><annotation encoding="application/x-tex">R-mod</annotation></semantics></math> and the composition endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mrow><mi>S</mi><mo>*</mo></mrow></msub><msubsup><mi>Q</mi> <mi>S</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">Q_{S*} Q^*_S</annotation></semantics></math> is underlying the corresponding <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>−</mo><mi>mod</mi></mrow><annotation encoding="application/x-tex">R-mod</annotation></semantics></math>. The component of the unit of its adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>R</mi></msub><mo>:</mo><mi>R</mi><mo>→</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi></mrow><annotation encoding="application/x-tex">\eta_R:R\to S^{-1} R</annotation></semantics></math> equals the canonical localization map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>M</mi></msub><mo>=</mo><mi>j</mi><msub><mo>⊗</mo> <mi>R</mi></msub><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>M</mi></msub></mrow><annotation encoding="application/x-tex">\eta_M = j\otimes_R\id_M</annotation></semantics></math>.</p> <h3 id="as_a_gabriel_localization">As a Gabriel localization</h3> <p>Given any multiplicative set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">S\subset R</annotation></semantics></math>, the set of all left ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">I\subset R</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\forall r\in R</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mi>R</mi><mo stretchy="false">|</mo><mi>z</mi><mi>r</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">}</mo><mo>∩</mo><mi>S</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\{z\in R| z r\in I\}\cap S\neq 0</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Gabriel+filter">Gabriel filter</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{F}_S</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is left Ore it is sufficient to ask that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∩</mo><mi>S</mi><mo>≠</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">I\cap S\neq\emptyset</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/Gabriel+localization">Gabriel localization</a> functor corresponding to this filter is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Q</mi> <mi>S</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">Q^*_S</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is left Ore.</p> <h2 id="properties">Properties</h2> <p>Basic property of Ore localization is flatness: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>R</mi></mrow><annotation encoding="application/x-tex">S^{-1}R</annotation></semantics></math> is a flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+domain">Ore domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization of a ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cohn+localization">Cohn localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+localization">noncommutative localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel+localization">Gabriel localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/calculus+of+fractions">calculus of fractions</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/K.+R.+Goodearl">K. R. Goodearl</a>, Robert B. Warfield, <em>An introduction to noncommutative Noetherian rings</em>, London Math. Soc. Student Texts <strong>16</strong> (1st ed,), 1989, xviii+303 pp.; or <strong>61</strong> (2nd ed.), 2004, xxiv+344 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zoran+%C5%A0koda">Zoran Škoda</a>, <em>Noncommutative localization in noncommutative geometry</em>, London Math. Society Lecture Note Series 330 (<a href="http://www.maths.ed.ac.uk/~aar/books/nlat.pdf">pdf</a>), ed. A. Ranicki; pp. 220–313, <a href="https://arxiv.org/abs/math.QA/0403276">math.QA/0403276</a>.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 15, 2024 at 15:37:36. 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