In this paper, we use the structure theory to prove an analog to a well-known theorem of Herstein as follows: Let R be a ring with center C such that for all x,y ? R either [x,y]= 0 or x-x [x,y]? C for some non negative integer n= n(x,y) dependingon x and y. Then R is commutative.

Let $R$ be a commutative Noetherian ring and let $fa$, $fb$ be two ideals of $R$ such that $R/({fa+fb})$ is Artinian. Let $M$, $N$ be two finitely generated $R$-modules. We prove that $H_{fb}^j(H_{fa}^t(M,N))$ is Artinian for $j=0,1$, where $t=inf{iin{mathbb{N}_0}: H_{fa}^i(M,N)$ is not finitelygenerated $}$. Also, we prove that if $DimSupp(H_{fa}^i(M,N))leq 2$, then $H_{fb}^1(H_{fa}^i(M,N))$ i...

let r be a commutative ring and (r) be its zerodivisor graph. in this article, we studywiener index and energy of γ(zn ) where n = pq or n = p2q and p, q are primes. a matlabcode for our calculations is also presented.

let $r$ be an arbitrary ring and $t$ be a submodule of an $r$-module $m$. a submodule $n$ of $m$ is called $t$-small in $m$ provided for each submodule $x$ of $m$, $tsubseteq x+n$ implies that $tsubseteq x$. we study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

in this paper, we use the structure theory to prove an analog to a well-known theorem of herstein as follows: let r be a ring with center c such that for all x,y ? r either [x,y]= 0 or x-x [x,y]? c for some non negative integer n= n(x,y) dependingon x and y. then r is commutative.

Let $R$ be an arbitrary ring and $T$ be a submodule of an $R$-module $M$. A submodule $N$ of $M$ is called $T$-small in $M$ provided for each submodule $X$ of $M$, $Tsubseteq X+N$ implies that $Tsubseteq X$. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper, we will introduce the secondary radical of a submodule $N$ of $M$ as the sum of all secondary submodules of $M$ contained in $N$, denoted by $sec^*(N)$, and explore the related properties. We will show that this class of modules contains the family of second radicals properly and can be regarded as a dual o...

let r be an associative ring with identity, c(r) be the category of com-plexes of r-modules and flat(c(r)) be the class of all at complexes of r-modules. we show that the at cotorsion theory (flat(c(r)); flat(c(r))−)have enough injectives in c(r). as an application, we prove that for each atcomplex f and each complex y of r-modules, exti (f,x)= 0, whenever ris n-perfect and i > n.

In the title compound, [Pd(C(37)H(55)P(2)Si)Cl], the Pd atom has a distorted square-planar geometry. The two five-membered rings adopt envelope conformations, while the four cyclo-hexane rings have chair conformations. The two planar aromatic rings are oriented at a dihedral angle of 28.79 (3)°.

The title compound, C(19)H(26)N(2)O(3), crystallizes with two independent mol-ecules in the asymmetric unit which differ in the twist of the phenyl rings with respect to the plane of the amide group [the C-C-C-O torsion angles are 121.5 (3) and -119.6 (3)° in the two mol-ecules. Both cyclo-hexane rings adopt chair conformations. In the crystal, weak C-H⋯O inter-actions occur. The crystal studie...