On simple-direct modules

Abstract

Recently, in a series of papers “simple” versions of direct-injective and direct-projective modules have been investigated. These modules are termed as “simple-direct-injective” and “simple-direct-projective,” respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.

Keywords:

• Simple-direct-projective
• Artinian rings
• H-rings
• perfect rings
• simple-direct-injective
• torsion submodule

2010 Mathematics Subject Classification:

• 16D50
• 16D60
• 18G25

Acknowledgments

The authors would like to thank the referee for his/her careful reading of the manuscript and for his/her valuable comments and suggestions.

Additional information

Funding

The third author is supported by TÜBITAK project number 119F176.