Dual Utumi modules

Abstract

A submodule N of a right R-module M is said to lie over a direct summand of M if, there is a decomposition $M=M_1\oplus M_2$ with $M_1\subseteq N$ and $N\cap M_2\stackrel{small}{,\hookrightarrow }{M}_2$. In this paper, a right R-module M is called a Dual-Utumi-Module ($DU$-module) if for any two proper submodules A and B of M with $M/A\cong M/B$ and A + B = M, there exist two summands K and L of M such that A lies over K, B lies over L and $K\cap L{\subseteq }^{\oplus }M.$ Dual-U-modules are strict generalizations of quasi-discrete, pseudo-discrete, and dual-square-free modules. In this paper several characterizations of $DU$-modules are provided which in turn are used to show that the class of $DU$-modules inherits some of the important features of the aforementioned classes of modules. For example, if $M=A\oplus B$ is a $DU$-module with $A\cong B,$ then M is quasi-projective and discrete. As an immediate application, a ring R is right perfect iff every quasi-projective right R-module is a $DU$-module. It is also shown that if M is a $DU$-module whose local summands are summands, then $M=Q\oplus P$ where Q and P are factor-orthogonal, Q is P-projective and dual–square-free, $Q={\oplus }_{\lambda \in \Lambda }{Q}_{\lambda }$ is a direct sum of pairwise non-isomorphic indecomposable modules, $P=C\oplus A\oplus B$ is quasi-projective and discrete with $A\cong B$, and C is isomorphic to a direct summand of $A\oplus B$. In particular, every quasi-discrete module has such a decomposition.

Keywords:

• Projective
• quasi-projective
• quasi-discrete
• U-modules
• dual-square-free and dual-automorphism-invariant modules
• clean rings and the exchange property

2010 Mathematics Subject Classification:

• Primary 16D40
• 16D50
• 16D60
• Secondary 16L30
• 16L60
• 16P20
• 16P40
• 16P60