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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Completely 0-simple semirings

Authors: Mireille Poinsignon Grillet and Pierre-Antoine Grillet
Journal: Trans. Amer. Math. Soc. 155 (1971), 19-33
MSC: Primary 16.96; Secondary 20.00
MathSciNet review: 0274531
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Abstract: A completely $ ( - 0)$ simple semiring is a semiring $ R$ which is $ ( - 0)$ simple and is the union of its $ ( - 0)$ minimal left ideals and the union of its $ ( - 0)$ minimal right ideals. Structure results are obtained for such semirings. First the multiplicative semigroup of $ R$ is completely $ ( - 0)$ simple; for any $ \mathcal{H}$-class $ H( \ne 0),H( \cup \{ 0\} )$ is a subsemiring. If furthermore $ R$ has a zero but is not a division ring, and if $ (H \cup \{ 0\} , + )$ has a completely simple kernel for some $ H$ as above (for instance, if $ R$ is compact or if the $ \mathcal{H}$-classes are finite), then (i) $ (R, + )$ is idempotent; (ii) $ R$ has no zero divisors, additively or multiplicatively. Additional results are given, concerning the additive $ \mathcal{J}$-classes of $ R$ and also $ ( - 0)$ minimal ideals of semirings in general.

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Keywords: Completely 0-simple semirings, completely simple semiring, minimal ideal, minimal left ideal, minimal right ideal, 0-minimal left ideal, 0-minimal right ideal, 0-minimal ideal, Green's relations of a semigroup, Green's relations of a semiring, division semiring, 0-division semiring, compact simple semiring, finite simple semiring, rectangular bands, bisimple semiring, 0-bisimple semiring, left translations of a semiring
Article copyright: © Copyright 1971 American Mathematical Society