The Jacobson radical of semigroup rings of commutative semigroups
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 \(J(R,S_ P,\Gamma ')=\{\sum a_{\alpha}\in J(R[S_ P])|\) \(a_{\alpha}\in R[(S_{\alpha})_ P]\}\). Let \({\mathbb{P}}\) be the set of all prime integers. The main result of the paper (theorem 3.10) states that, for S separative \[ J(R[S])=J_ 1(R)[S]+\sum_{p\in {\mathbb{P}}}I(J_{1,p}(R),S,\xi_ p)+J(R,S_ P,\Gamma '), \] provided \(J_ 1(R)=J_{n+1}(R)\) for all positive integers n. It is known that \(J_ 1(R)=J_{n+1}(R)\) for all positive n if R satisfies a polynomial identity, therefore the assertion of Theorem 3.10 holds for R commutative.
scientific article; zbMATH DE number 4000230
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