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
Abstract: A completely simple semiring is a semiring which is simple and is the union of its minimal left ideals and the union of its minimal right ideals. Structure results are obtained for such semirings. First the multiplicative semigroup of is completely simple; for any -class is a subsemiring. If furthermore has a zero but is not a division ring, and if has a completely simple kernel for some as above (for instance, if is compact or if the -classes are finite), then (i) is idempotent; (ii) has no zero divisors, additively or multiplicatively. Additional results are given, concerning the additive -classes of and also minimal ideals of semirings in general.
-  Samuel Bourne, On multiplicative idempotents of a potent semiring, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 632–638. MR 0080085
-  A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
-  Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR 0222106
-  Gérard Lallement, Demi-groupes réguliers, Ann. Mat. Pura Appl. (4) 77 (1967), 47–129 (French). MR 0225915, https://doi.org/10.1007/BF02416940
-  M. O. Poinsignon [ = M. P. Grillet], Green's relations in a semiring, Portugal. Math. (to appear).
- S. Bourne, On the multiplicative idempotents of a potent semiring, Proc. Nat. Acad Sci. U.S.A. 42 (1956), 632-638. MR 18, 188. MR 0080085 (18:188b)
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vols. I, II, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961, 1967. MR 24 #A2627; MR 36 #1558. MR 0132791 (24:A2627)
- N. Jacobson, Structure of rings, 2nd rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1964. MR 36 #5158. MR 0222106 (36:5158)
- G. Lallement, Demi-groupes réguliers, Ann. Mat. Pura Appl. (4) 77 (1967), 47-129. MR 37 #1505. MR 0225915 (37:1505)
- M. O. Poinsignon [ = M. P. Grillet], Green's relations in a semiring, Portugal. Math. (to appear).
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