×

Identities of semigroup rings over semilattices of completely (\(0\)- )simple semigroups. (English) Zbl 0835.16023

In what follows \(R\) is an associative ring, \(S\) is a semigroup, \(E(S)\) is the set of all idempotents of \(S\), \(\langle X \rangle\) is the subsemigroup of \(S\) generated by \(X \subseteq S\), \(RS\) is the contracted semigroup ring of \(S\) over \(R\). In [F. Li, Semigroup Forum 46, 27- 31 (1993; Zbl 0787.16024)] the following interesting problem is raised: suppose that \(S\) is a regular semigroup such that \(RS\) is a ring with identity; does it mean that \(R \langle E(S) \rangle\) is also a ring with identity (equivalently: does the identity of \(RS\) belong to \(R \langle E(S) \rangle)\)? In 1969 Wenger showed that, for \(S\) inverse, the answer is in the affirmative. In a previous paper the author proved that, for \(S\) orthodox, the problem mentioned also has a positive solution. In the present paper the same is proved for \(S\) a semilattice of completely \(0\)- simple and completely simple semigroups, in particular, for any \(S\) completely regular.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
20M25 Semigroup rings, multiplicative semigroups of rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
20M17 Regular semigroups

Citations:

Zbl 0787.16024
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Clifford, A. H. and G. B. Preston,The Algebraic Theory of Semigroups, Vol. I (1961) and Vol. II (1967), Math. Surveys of Amer. Math. Soc. 7. · Zbl 0111.03403
[2] Howie, J. M.,An Introduction to Semigroup Theory, 1976, Academic Press, London, New York, San Francisco. · Zbl 0355.20056
[3] Li, Fang,The Existence of Identity of Orthodox Semigroup Rings, Semigroup Forum46 (1993), 27–31. · Zbl 0787.16024 · doi:10.1007/BF02573540
[4] Petrich, M.,Lectures in Semigroups, 1977, Wiley, London. · Zbl 0369.20036
[5] Ponizovskiî, J. S.,Semigroup Rings, Semigroup Forum36 (1987), 1–46. · Zbl 0629.20039 · doi:10.1007/BF02575003
[6] Song, G. T.,Identities of Orthodox Semigroup Rings, Semigroup Forum49 (1994), 239–246. · Zbl 0815.20065 · doi:10.1007/BF02573486
[7] Wenger, R. H.,Self-Injective Semigroup Rings for Finite Inverse Semigroups Proc. Amer. Math. Soc.20 (1969), 213–216. · Zbl 0167.02004 · doi:10.1090/S0002-9939-1969-0232869-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.