Modules with ascending chain condition on annihilators and Goldie modules

ABSTRACT

Using the concepts of prime module, semiprime module and the concept of ascending chain condition (ACC) on annihilators for an R-module M, we prove that if M is semiprime and projective in σ[M], such that M satisfies ACC on annihilators, then M has finitely many minimal prime submodules. Moreover, if each submodule N⊆M contains a uniform submodule, we prove that there is a bijective correspondence between a complete set of representatives of isomorphism classes of indecomposable non-M-singular injective modules in σ[M] and the set of minimal primes in M. If M is a Goldie module, then $\hat{M}\cong E_1^{k_1}\oplus E_2^{k_2}\oplus \cdots \oplus E_n^{k_n}$, where each E_i is a uniform M-injective module. As an application, new characterizations of left Goldie rings are obtained.

KEYWORDS:

• Goldie modules
• indecomposable modules
• prime modules
• semiprime modules
• torsion theory

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S90
• 16D50
• 16P50
• 16P70

Acknowledgment

The authors want to thank the referee for the carefully reading and the useful comments to improve this paper.