A ring $R$ is called a left APP-ring if the left annihilator $l_R\left(Ra\right)$ is right $s$-unital as an ideal of $R$ for any element $a\in R$. We consider left APP-property of the skew formal power series ring $R\left[\right[x;\alpha \left]\right]$ where $\alpha$ is a ring automorphism of $R$. It is shown that if $R$ is a ring satisfying descending chain condition on right annihilators then $R\left[\right[x;\alpha \left]\right]$ is left APP if and only if for any sequence $(b_0,b_1,\cdots )$ of elements of $R$ the ideal $l_R$ $\left({\sum }_{j=0}^{\infty }{\sum }_{k=0}^{\infty }R{\alpha }^k\left(b_j\right)\right)$ is right $s$-unital. As an application we give a sufficient condition under which...

Let $R$ be an associative ring with identity and let $J\left(R\right)$ denote the Jacobson radical of $R$. $R$ is said to be semilocal if $R/J\left(R\right)$ is Artinian. In this paper we give necessary and sufficient conditions for the group ring $RG$, where $G$ is an abelian group, to be semilocal.

It is shown that a ring $R$ is a $GM$-ring if and only if there exists a complete orthogonal set $\{e_1,\cdots ,e_n\}$ of idempotents such that all $e_{i}R{e}_i$ are $GM$-rings. We also investigate $GM$-rings for Morita contexts, module extensions and power series rings.

Let $m\ge 0,r\ge 0,s\ge 0,q\ge 0$ be fixed integers. Suppose that $R$ is an associative ring with unity $1$ in which for each $x,y\in R$ there exist polynomials $f\left(X\right)\in X^{2}ZZ\left[X\right],g\left(X\right),h\left(X\right)\in XZZ\left[X\right]$ such that $\{1-g\left(y{x}^m\right)\}[x,x^{r}y-x^{s}f\left(y{x}^m\right)x^q]\{1-h\left(y{x}^m\right)\}=0$. Then $R$ is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of $x$ and $y$. Finally, commutativity of one sided s-unital ring is also obtained when $R$ satisfies some related ring properties.

A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $\left({\sum }_{i\ge 0}{a}_i{x}^i\right)\left({\sum }_{j\ge 0}{b}_j{x}^j\right)=0$ in R[x] (resp., in R[[x]]), then $a_i{b}_j=0$ for all i and j. This paper deals with a unified generalization of the two concepts (see Definition 2). Some known results on Armendariz rings are extended to this more general situation and new results are obtained as consequences. For instance, it is proved that a ring R is Armendariz of power series type iff the same is true of R[[x]]. For an injective endomorphism...

Let $m>1,s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p=p\left(x\right)\ge 0,q=q\left(x\right)\ge 0,n=n\left(x\right)\ge 0,r=r\left(x\right)\ge 0$ such that either $x^p[x^n,y]x^q=x^r[x,y^m]y^s$ or $x^p[x^n,y]x^q=y^s[x,y^m]x^r$ for all $y\in R$. In the present paper it is shown that $R$ is commutative if it satisfies the property $Q\left(m\right)$ (i.e. for all $x,y\in R,m[x,y]=0$ implies $[x,y]=0$).