Completely $0$-simple semirings

Mireille Poinsignon Grillet, Pierre-Antoine Grillet
1971 Transactions of the American Mathematical Society  
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 Jf-class 7/(^0), H (u {0}) is a subsemiring. If furthermore R has a zero but is not a division ring, and if (H <J {0}, +) has a completely simple kernel for some H as above (for instance, if R is compact
more » ... , if R is compact or if the ^-classes are finite), then (i) {R, + ) is idempotent ; (ii) R has no zero divisors, additively or multiplicatively. Additional results are given, concerning the additive ./-classes of R and also (0-) minimal ideals of semirings in general. Received by the editors December 22, 1969. AMS 1969 subject classifications. Primary 1696. Key words and phrases. 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.
doi:10.1090/s0002-9947-1971-0274531-8 fatcat:zukjnknxejfvteoypj2vdebcgi