فهرست مطالب y. talebi

‎We introduce the notion of lifting Baer modules‎, ‎as a generalization of both Baer and lifting modules and give some of their properties‎. ‎A module $M$ is called lifting Baer if right annihilator of a left ideal of ${\rm End}(M)$ lies above a direct summand of M‎. ‎Also‎, ‎we define the concepts of $r$-supplemented and amply $r$-supplemented modules‎. ‎It is shown that an amply $r$-supplemened module M that every supplement submodule‎, ‎is a direct summand of $M$‎, ‎is lifting Baer‎. ‎The relationships between Baer modules and lifting Baer modules are investigated.‎ ‎Morever‎, ‎we prove that the endomorphism ring of any lifting Baer module is‎ ‎lifting Baer ring.‎

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 $tCC$.

Let R be a commutative ring with identity and M an R-module. The Scalar-Product Graph of M is defined as the graph GR(M) with the vertex set M and two distinct vertices x and y are adjacent if and only if there exist r or s belong to R such that x = ry or y = sx. In this paper , we discuss connectivity and planarity of these graphs and computing diameter and girth of GR(M). Also we show some of these graphs is weakly perfect.

Let M be a right R-module. We call M Rad-H-supplemented if for each Y ≤ M there exists a direct summand D of M such that (Y +D)/D ⊆ (Rad(M)+D)/D and (Y +D)/Y ⊆ (Rad(M)+Y )/Y . It is shown that: (1) Let M = M1 ⊕ M2, where M1 is a fully invariant submodule of M. If M is Rad-H-supplemented, then M1 and M2 are Rad-Hsupplemented. (2) Let M = M1 ⊕M2 be a duo module and Rad-⊕- supplemented. If M1 is radical M2-sejective (or M2 is radical M1- sejective), then M is Rad-H-supplemented. (3) Let M = ⊕ n i=1Mi be a finite direct sum of modules. If Mi is generalized radical Mj - projective for all j > i and each Mi is Rad-H-supplemented, then M is Rad-H-supplemented.