Journal of Algebra and Related Topics
Volume:8 Issue: 1, Spring 2020

In this paper we introduce the concept of $tCC$-modu-les which is a proper generalizationof ($t$-)lifting modules. Let $M$ be a module over a ring $R$.We call $M$ a $tCC$-module(related to $t$-coclosed submodules) provided that for every$t$-coclosed submodule $N$ of $M$, there exists a direct summand $K$ of $M$such that $M=N+K$ and $Ncap Kll K$.We prove that a module with $(D_3)$ property is $tCC$if and only if every direct summand of $M$ is $tCC$. It is also shownthat an amply supplemented module $M$ is $tCC$ if and only if $M$ decomposed to$overline{Z}^2(M)$ and a submodule $L$ of $M$ that both of them are $tCC$.

We study the pullback Lie algebra (group) bundle of a Lie algebra (group) bundle and show that the Lie algebra bundle of the pullback of a Lie group bundle $mathfrak{G}$ is isomorphic to the pullback of the Lie algebra bundle of $mathfrak{G}$. Then, using the notion of Lie connection on a Lie algebra bundle, we show that the pullbacks of a Lie algebra bundle $xi$ over a smooth manifold $M$ with respect to two smooth homotopic functions $f_0 , f_1 : N rightarrow M$ are isomorphic to Lie algebra bundles over $N$.

Let $R$ be a commutative ring and $M$ be an $R$-module. The intersection graph of annihilatorsubmodules of $M$, denoted by ${GA(M)}$, is a simple undirected graph whose vertices are the classes of elements of $Z(M)setminus {rm Ann}_R(M)$ and two distinct classes $[a]$ and$[b]$ are adjacent if and only if ${rm Ann}_M(a)cap {rm Ann}_M(b)not=0$. In this paper, we studythe diameter and girth of $overline{GA(M)}$. Furthermore, we calculate the domination number,metric dimension, adjacency metric dimension and local metric dimension of $overline{GA(M)}$.

Let $G$ be a group with identity $e$. Let $R$ be an associative $G$-graded ring and $M$ be a $G$-graded $R$-module. In this article, we intruduce the concept of graded 2-absorbing submodules as a generalization of graded prime submodules over non-commutative graded rings. Moreover, we get some properties of such graded submodules.

The question of identifying the elements of the center of a nearring and of determining when that center is a subnearring is an area of continued research. We consider the centers of centralizer nearrings, MI(Sn), determined by the symmetric groups Sn with n≥3 and the inner automorphisms I=Inn Sn. General tools for determining elements of the center of MI(Sn) are developed, and we use these to list the specific elements in the centers of MI(S4), MI(S5), and MI(S6).

Providing a description of linked ideals in a commutative Noetherian ring in terms of some associated prime ideals, we make a characterization of Cohen-Macaulay, Gorenstein and regular local rings in terms of their linked ideals. More precisely, it is shown that the local ring (R,\fm) is Cohen-Macaulay if and only if any linked ideal is unmixed. Also, (R,\fm) is Gorenstein if and only if any unmixed ideal \fa is linked by every maximal regular sequence in \fa. We also compute the annihilator of top local cohomology modules in some special cases.