Modules Whose Submodules are Essentially Embedded in Direct Summands

Abstract

A module M is said to satisfy the C ₁₂ condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C ₁₁ (and hence also C ₁) condition implies the C ₁₂ condition. We show that the class of C ₁₂-modules is closed under direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a -module with cancellable socle and satisfies ascending chain (respectively, descending chain) condition on essential submodules, then M is a direct sum of a semisimple and a Noetherian (respectively, Artinian) submodules. Moreover, a C ₁₂-module with cancellable socle is shown to be a direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.

Key Words:

• Cancellable module
• Extending module
• Finite uniform dimension
• C ₁₁-Module
• C ₁₂-Module

2000 Mathematics Subject Classification:

• Primary 16D70
• Secondary 16D50
• 16D40

Notes

Communicated by T. Albu.