On secondary and representable modules over almost Dedekind domains

Abstract

Matlis showed that an injective module over a commutative Noetherian ring R can be completely decomposed as a direct sum of indecomposable injective submodules. In this paper, we prove the Matlis’ Theorem for almost Dedekind domains. Then we characterize the secondary modules and classify the indecomposable secondary modules over almost Dedekind domains. Also we prove every P-secondary module over an almost Dedekind domain is pure-injective, where $P\ne 0.$ Finally, we characterize the representable finitely generated modules over almost Dedekind domains.

Keywords:

• Almost Dedekind domains
• secondary modules
• representable modules
• FP-injective modules
• finitely injective modules
• divisible modules

2010 Mathematics Subject Classification:

• 13C11
• 13C99
• 13F05

Acknowledgment

The authors would like to thank the referee for his/her useful suggestions that improved the presentation of this paper.