Simple-direct-modules

ABSTRACT

A right R-module M is called simple-direct-injective if, whenever, A and B are simple submodules of M with A≅B, and B⊆^⊕M, then A⊆^⊕M. Dually, M is called simple-direct-projective if, whenever, A and B are submodules of M with M∕A≅B⊆^⊕M and B simple, then A⊆^⊕M. In this paper, we continue our investigation of these classes of modules strengthening many of the established results on the subject. For example, we show that a ring R is uniserial (artinian serial) with J²(R) = 0 iff every simple-direct-projective right R-module is an SSP-module (SIP-module) iff every simple-direct-injective right R-module is an SIP-module (SSP-module).

KEYWORDS:

• Artinian serial rings and uniserial rings
• C3- and D3-modules
• CS and lifting modules
• simple-direct-injective and simple-direct-projective modules
• SSP and SIP modules

1991 MATHEMATICS SUBJECT CLASSIFICATION:

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

Acknowledgment

Part of this work was carried out while the fourth author was visiting the Mathematics Department of Cairo University during the Spring of 2016. The author would like to take this opportunity to thank members of the Mathematics Department at Cairo University for their kind hospitality and warm reception. The fourth author acknowledges support from the OSU Mathematics Research Institute.