The Jacobson radical of semigroup rings of commutative semigroups
type: Item
instance of: scholarly article
title: The Jacobson radical of semigroup rings of commutative semigroups
zbMATH Open document ID: 0617.20040
DOI: 10.1016/0021-8693(87)90175-X
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Recently \textit{W. D. Munn} [Math. Proc. Camb. Philos. Soc. 93, 237-246 (1983; Zbl 0528.20053)] described the Jacobson radical of a semigroup algebra F[S], where F is a field and S is a commutative semigroup. Earlier \textit{R. Gilmer} and \textit{T. Parker} [Mich. Math. J. 22, 97-108 (1975; Zbl 0301.20064)] described the nil radical of a commutative semigroup ring (another description of that radical is given by \textit{W. D. Munn} [Proc. R. Soc. Edinb., Sect. A 99, 387-398 (1985; Zbl 0565.20048)]. In the paper under review the next step is made - a description of the Jacobson radical of a commutative semigroup ring is given. Let S be a commutative semigroup, and let R be an associative ring (not necessarily commutative). If \(\rho\) is a congruence on S, then \(I(R,S,\rho)=\{\sum (rs-rt)|\) \(r\in R\), (s,t)\(\in \rho \}\). For any ring A, J(A) denotes the Jacobson radical of A. If A is an ideal of a ring B, then \(A_ n=\{x\in B|\) xn\(\in A\}\). Let n, m be positive integers; then \[ J_ n(R)=R\cap J(R[x_ 1,...,x_ n],\quad J_{n,m}(R)=(J_ n(R))_ m. \] A semigroup S is separative (p-separative, p a prime integer) iff the quasi-identity \([x^ 2=xy=y^ 2\Rightarrow x=y]\) (the quasi-identity \(x^ p=y^ p\Rightarrow x=y]\) holds on S. A congruence \(\rho\) on S is separative (p-separative) iff S/\(\rho\) is separative (p-separative). By \(\xi\) (resp. \(\xi_ p)\) is denoted the least separative (p-separative) congruence on S. \(\xi\) gives rise to an epimorphism R[S]\(\to R[S/\xi]\). Since the kernel of that mapping is I(R,S,\(\xi)\) (and hence is known) it is enough to describe J(R[S]) for S separative. So assume that S is separative. Then \(S=\cup S_{\alpha}\) (\(\alpha\in \Gamma)\) is a semilattice of cancellative semigroups \(S_{\alpha}\) (\(\alpha\in \Gamma)\). If this is the case, \(\Gamma\) ' is a subset of \(\Gamma\) : \(\alpha\in \Gamma '\) iff \(S_{\alpha}\) is not periodic. Let \(S_ P\) be a maximal periodic subsemigroup of S, and let \(J(R,S_ P,\Gamma ')=\{\…
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 16Nxx
zbMATH DE Number: 4000230
zbMATH Keywords: Jacobson radical
zbMATH Keywords: semigroup algebra
zbMATH Keywords: commutative semigroup
zbMATH Keywords: nil radical
zbMATH Keywords: commutative semigroup ring
zbMATH Keywords: quasi-identity
zbMATH Keywords: separative
zbMATH Keywords: p-separative
zbMATH Keywords: congruence
zbMATH Keywords: semilattice of cancellative semigroups
zbMATH Keywords: polynomial identity
author: Eric Jespers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4000230