On V-semirings and Semirings All of Whose Cyclic Semimodules Are Injective

Abstract

In this article, we introduce and study V- and CI-semirings—semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and antibounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple antibounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.

Key Words:

• CI-semirings
• Congruence-simple semirings
• Injective semimodules
• Morita equivalence of semirings
• Simple semimodules
• Semisimple semirings
• V-semirings

2010 Mathematics Subject Classification:

• Primary: 16Y60
• 16D99
• 06A12
• Secondary: 18A40
• 18G05
• 20M18

Notes

Communicated by V. A. Artamonov.

Dedicated to the memory of Professor L. A. Skornjakov.