On V-semirings and Semirings All of Whose Cyclic Semimodules Are Injective |
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Authors: | J Y Abuhlail Y Katsov T G Nam |
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Institution: | 1. Department of Mathematics and Statistics , King Fahd University of Petroleum and Minerals , Dhahran , Saudi Arabia;2. Department of Mathematics , Hanover College , Hanover , Indiana , USA;3. Institute of Mathematics , Hanoi , Vietnam |
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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. |
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Keywords: | CI-semirings Congruence-simple semirings Injective semimodules Morita equivalence of semirings Simple semimodules Semisimple semirings V-semirings |
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