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Search: MSC category 16R50 ( Other kinds of identities (generalized polynomial, rational, involution) )

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1. CMB 2012 (vol 56 pp. 584)

Liau, Pao-Kuei; Liu, Cheng-Kai
On Automorphisms and Commutativity in Semiprime Rings
Let $R$ be a semiprime ring with center $Z(R)$. For $x,y\in R$, we denote by $[x,y]=xy-yx$ the commutator of $x$ and $y$. If $\sigma$ is a non-identity automorphism of $R$ such that $$ \Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0 $$ for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$ such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when $R$ is a prime ring, $R$ is commutative.

Keywords:automorphism, generalized polynomial identity (GPI)
Categories:16N60, 16W20, 16R50

2. CMB 2011 (vol 55 pp. 271)

Di Vincenzo, M. Onofrio; Nardozza, Vincenzo
On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras
Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.

Keywords:exponent, polynomial identities, graded algebras
Categories:16R50, 16R10, 16W50

3. CMB 2009 (vol 52 pp. 267)

Ko\c{s}an, Muhammet Tamer
Extensions of Rings Having McCoy Condition
Let $R$ be an associative ring with unity. Then $R$ is said to be a {\it right McCoy ring} when the equation $f(x)g(x)=0$ (over $R[x]$), where $0\neq f(x),g(x) \in R[x]$, implies that there exists a nonzero element $c\in R$ such that $f(x)c=0$. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if $R$ is a right McCoy ring, then $R[x]/(x^n)$ is a right McCoy ring for any positive integer $n\geq 2$ .

Keywords:right McCoy ring, Armendariz ring, reduced ring, reversible ring, semicommutative ring
Categories:16D10, 16D80, 16R50

4. CMB 2003 (vol 46 pp. 14)

Bahturin, Yu. A.; Parmenter, M. M.
Generalized Commutativity in Group Algebras
We study group algebras $FG$ which can be graded by a finite abelian group $\Gamma$ such that $FG$ is $\beta$-commutative for a skew-symmetric bicharacter $\beta$ on $\Gamma$ with values in $F^*$.

Categories:16S34, 16R50, 16U80, 16W10, 16W55

5. CMB 1998 (vol 41 pp. 118)

Valenti, Angela
On permanental identities of symmetric and skew-symmetric matrices in characteristic \lowercase{$p$}
Let $M_n(F)$ be the algebra of $n \times n$ matrices over a field $F$ of characteristic $p>2$ and let $\ast$ be an involution on $M_n(F)$. If $s_1, \ldots, s_r$ are symmetric variables we determine the smallest $r$ such that the polynomial $$ P_{r}(s_1, \ldots, s_{r}) = \sum_{\sigma \in {\cal S}_r}s_{\sigma(1)}\cdots s_{\sigma(r)} $$ is a $\ast$-polynomial identity of $M_n(F)$ under either the symplectic or the transpose involution. We also prove an analogous result for the polynomial $$ C_r(k_1, \ldots, k_r, k'_1, \ldots, k'_r) = \sum_ {\sigma, \tau \in {\cal S}_r}k_{\sigma(1)}k'_{\tau(1)}\cdots k_{\sigma(r)}k'_{\tau(r)} $$ where $k_1, \ldots, k_r, k'_1, \ldots, k'_r$ are skew variables under the transpose involution.


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