<?xml version="1.0" encoding="utf-8"?> <feed xmlns="http://www.w3.org/2005/Atom"> <title type="text">Recent zbMATH articles in MSC 16W10</title> <id>https://zbmath.org/atom/cc/16W10</id> <updated>2025-03-04T19:44:26.967317Z</updated> <link href="https://zbmath.org/" /> <link href="https://zbmath.org/atom/cc/16W10" rel="self" /> <generator>Werkzeug</generator> <entry xml:base="https://zbmath.org/atom/cc/16W10"> <title type="text">Rings close to periodic with applications to matrix, endomorphism and group rings</title> <id>https://zbmath.org/1551.16024</id> <updated>2025-03-04T19:44:26.967317Z</updated> <link href="https://zbmath.org/1551.16024" /> <author> <name>&quot;Abyzov, Adel N.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:abyzov.adel-nailevich</uri> </author> <author> <name>&quot;Barati, Ruhollah&quot;</name> <uri>https://zbmath.org/authors/?q=ai:barati.ruhollah</uri> </author> <author> <name>&quot;Danchev, Peter V.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:danchev.peter-vassilev</uri> </author> <content type="text">A ring \(R\) is called periodic if, for every \(a\) in \(R\), there exist two distinct positive integers \(m\) and \(n\) such that \(a^m=a^n\). The paper is devoted to the search for conditions and criteria under which various types of rings are periodic. The authors consider group rings, matrix rings, endomorphism rings, and also the tensor products of algebras. In particular, it is proved that if \(R\) is a right (resp., left) perfect periodic ring and \(G\) is a locally finite group then the group ring \(RG\) is periodic (Theorem 1.6) as well as the matrix ring \(M_n(R)\) is periodic for all \(n\geq 1\) (Theorem 2.10). The main result of the paper on endomorphism rings (Theorem 3.5) states that for any abelian group \(G\), the endomorphism ring \(E(G)\) is periodic if and only if the group \(G\) is finite. The authors also prove some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic (Theorems 4.2 and 4.4). In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated (Section 5). Reviewer: Anatolii Tushev (Dnipro)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/16W10"> <title type="text">Generalized Baer \(\ast \)-rings</title> <id>https://zbmath.org/1551.16035</id> <updated>2025-03-04T19:44:26.967317Z</updated> <link href="https://zbmath.org/1551.16035" /> <author> <name>&quot;Ahmadi, M.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ahmadi.morteza|ahmadi.mahnaz|ahmadi.mandana|ahmadi.mosayeb|ahmadi.mohamadreza|ahmadi.m-yar|ahmadi.mohsen|ahmadi.masoud|ahmadi.mohammad-taghi|ahmadi.mansour|ahmadi.mohammad-reza|ahmadi.mohammad-hossein|ahmadi.mohammad-vali|ahmadi.mojtaba-pour|ahmadi.marzieh|ahmadi.m-barkhordari|ahmadi.masumeh|ahmadi.mohammad-javad|ahmadi.maryam-khan|ahmadi.mohammad-bagher|ahmadi.mahmood|ahmadi.mohamad|ahmadi.mehdi|faghih-ahmadi.masoumeh|ahmadi.mahdi|ahmadi.mohammad-mahdi|ahmadi.majid|ahmadi.muhammad|ahmadi.mazda</uri> </author> <author> <name>&quot;Moussavi, A.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:moussavi.ahmad</uri> </author> <content type="text">Summary: We say that a \(\ast \)-ring \(R\) is a generalized Baer \(\ast \)-ring if, for each nonempty subset \(S\) of \(R \), the right annihilator \(r_R(S^n)\) is generated as a right ideal by a projection for some positive integer \(n\) depending on \(S \). Each nonempty set of projections in a generalized Baer \(\ast \)-ring is a complete lattice. We study the properties of the \(\ast \)-rings. We show that abelian generalized Baer \(\ast \)-rings are well behaved with respect to finite direct products and certain triangular matrix extensions. We give some algebraic examples that are generalized Baer \(\ast \)-rings but not Baer \(\ast \)-rings. We obtain the classes of both finite and infinite dimensional Banach \(\ast \)-algebras which are generalized Baer \(\ast \)-rings but not Baer \(\ast \)-rings. We define a generalized \(AW^* \)-algebra as a \(C^* \)-algebra that is a generalized Baer \(\ast \)-ring. The concept of generalized \(AW^* \)-algebra is a generalization of \(AW^* \)-algebra, an algebraic extension of a \(W^* \)-algebra. We show that for semicommutative \(C^* \)-algebras the notions of generalized \(AW^* \)-algebra and \(AW^* \)-algebra coincide.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/16W10"> <title type="text">Values of polynomials on centrally closed prime algebras</title> <id>https://zbmath.org/1551.16036</id> <updated>2025-03-04T19:44:26.967317Z</updated> <link href="https://zbmath.org/1551.16036" /> <author> <name>&quot;Lee, Tsiu-Kwen&quot;</name> <uri>https://zbmath.org/authors/?q=ai:lee.tsiu-kwen</uri> </author> <author> <name>&quot;Lin, Jheng-Huei&quot;</name> <uri>https://zbmath.org/authors/?q=ai:lin.jheng-huei</uri> </author> <content type="text">Let \(R\) be a simple algebra over its extended centroid \(C\), and let \(f(X_1,\dots , X_n)\) be a noncommutative polynomial having zero constant term. Denote by \(f(R)^+\) the additive subgroup of \(R\) generated by all elements \(f(x_1, \ldots , x_n)\) for \(x_i \in R\). The authors prove that if \(\mathrm{char}\ R = 0\), then \(f(R)^+\) is equal to \(\{0\}\), \(C\), \([R, R]\), or \(R\). In the case \(\mathrm{char}\ R = p &gt; 0\), the authors provide an example of a polynomial \(f\) satisfying \([R, R]\subsetneq f(R)^+ \subsetneq R\). Also, the authors characterize polynomials \(f\) with \(f(R)^+ = [R, R]\) in the case \(\mathrm{char}\ R = 0\) and \(R \ne [R, R]\). Moreover, the authors work with centrally closed prime algebras to get more general results. Reviewer: Victor Petrogradsky (Bras铆lia)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/16W10"> <title type="text">Lie triple centralizers on generalized matrix algebras</title> <id>https://zbmath.org/1551.16040</id> <updated>2025-03-04T19:44:26.967317Z</updated> <link href="https://zbmath.org/1551.16040" /> <author> <name>&quot;Fadaee, Behrooz&quot;</name> <uri>https://zbmath.org/authors/?q=ai:fadaee.behrooz</uri> </author> <author> <name>&quot;Ghahramani, Hoger&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ghahramani.hoger</uri> </author> <author> <name>&quot;Jing, Wu&quot;</name> <uri>https://zbmath.org/authors/?q=ai:jing.wu</uri> </author> <content type="text">In this paper, the authors introduce the concept of a Lie triple centralizer on generalized matrix algebras. A linear map \(\varphi: A \to A\) is defined as a Lie triple centralizer if it satisfies \(\varphi([[a,b],c]) = [[\varphi(a),b],c]\) for all \(a, b, c \in A\). Below are the key theorems presented in the paper: Definitions of symbols: \begin{itemize} \item \(U\) denotes a generalized matrix algebra, defined as \(U = \begin{bmatrix} A &amp; M \\ N &amp; B \end{bmatrix}\), where \(A\) and \(B\) are algebras, \(M\) is an \((A,B)\)-bimodule, and \(N\) is a \((B,A)\)-bimodule. \item \(Z(A)\) denotes the center of an algebra \(A\), defined as \(Z(A) = \{a \in A : ac = ca \text{ for all } c \in A\}\). \item \(Z(U)\) denotes the center of the algebra \(U\), with \(Z(U) = \begin{bmatrix} Z(A) &amp; 0 \\ 0 &amp; Z(B) \end{bmatrix}\) under certain conditions. \item \(M\) and \(N\) are bimodules that satisfy the conditions: if \(a \in A\), \(aM = 0\), and \(Na = 0\), then \(a = 0\). Similarly, if \(b \in B\), \(Mb = 0\), and \(bN = 0\), then \(b = 0\). \item \(\pi_A\) and \(\pi_B\) are projection mappings defined as \(\pi_A\left(\begin{bmatrix} a &amp; m \\ n &amp; b \end{bmatrix}\right) = a\) and \(\pi_B\left(\begin{bmatrix} a &amp; m \\ n &amp; b \end{bmatrix}\right) = b\). \end{itemize} Main theorems: \begin{itemize} \item Theorem 3.1: A linear map \(\varphi: U \to U\) is a Lie triple centralizer if and only if \(\varphi\) has the form: \[ \varphi \left( \begin{bmatrix} a &amp; m \\ n &amp; b \end{bmatrix} \right) = \begin{bmatrix} \alpha_1(a) + \beta_1(b) &amp; \tau_2(m) \\ \gamma_3(n) &amp; \alpha_4(a) + \beta_4(b) \end{bmatrix} \] where \(\alpha_1 : A \to A\), \(\beta_1 : B \to [A,A]'\), \(\tau_2 : M \to M\), \(\gamma_3 : N \to N\), \(\alpha_4 : A \to [B,B]'\), and \(\beta_4 : B \to B\) are linear mappings satisfying the following conditions: \begin{itemize} \item[(1)] \(\alpha_1\) is a Lie triple centralizer on \(A\), \(\alpha_4([[a_1, a_2], a_3]) = 0\), and \(\alpha_1(mn) - \beta_1(nm) = \tau_2(m)n = m\gamma_3(n)\) for all \(m \in M\) and \(n \in N\). \item[(2)] \(\beta_4\) is a Lie triple centralizer on \(B\), \(\beta_1([[b_1, b_2], b_3]) = 0\), and \(\alpha_4(nm) - \beta_4(mn) = n\tau_2(m) = \gamma_3(n)m\) for all \(m \in M\) and \(n \in N\). \item[(3)] \(\tau_2(am) = a\tau_2(m) = \alpha_1(a)m - m\alpha_4(a)\) and \(\tau_2(mb) = \tau_2(m)b = m\beta_4(b) - \beta_1(b)m\) for any \(a \in A\), \(b \in B\), and \(m \in M\). \item[(4)] \(\gamma_3(na) = \gamma_3(n)a = n\alpha_1(a) - \alpha_4(a)n\) and \(\gamma_3(bn) = b\gamma_3(n) = \beta_4(b)n - n\beta_1(b)\) for any \(a \in A\), \(b \in B\), and \(n \in N\). \end{itemize} \item Theorem 3.3: Let \(U\) satisfy the conditions: \begin{align*} &amp; a \in A, aM = 0 \text{ and } Na = 0 \Rightarrow a = 0, \\ &amp; b \in B, Mb = 0 \text{ and } bN = 0 \Rightarrow b = 0. \end{align*} A linear map \(\varphi : U \to U\) is a proper Lie triple centralizer if and only if \(\alpha_4(A) \subseteq \pi_B(Z(U))\) and \(\beta_1(B) \subseteq \pi_A(Z(U))\). \item Corollary 3.4: Under the same conditions as Theorem 3.3, if either \(\pi_B(Z(U)) = Z(B)\) or \([[A,A],A] = A\) holds, then any Lie triple centralizer on \(U\) is a proper Lie triple centralizer. \item Theorem 4.2: Let \(U\) satisfy the conditions stated in Theorem 3.3. If \(\Lambda: U \to U\) is a generalized Lie triple derivation associated with a Lie triple derivation \(\xi: U \to U\), then \(\Lambda(A) = \delta(A) + d(A) + \psi(A) + \lambda A\) for any \(A \in U\), where \(\delta\) is a derivation on \(U\), \(d\) is a singular Jordan derivation on \(U\), \(\lambda \in Z(U)\), and \(\psi: U \to Z(U)\) is a linear map that vanishes on \([[U, U], U]\). \end{itemize} These results extend the theory of Lie triple centralizers and provide a foundation for further studies in the Lie structure of algebras. Reviewer: Hafedh Alnoghashi (Amran)</content> </entry> </feed>