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A description of the Jacobson radical of semigroup rings of commutative semigroups. (English) Zbl 0599.20103

Group and semigroup rings, Proc. Int. Conf., Johannesburg/South Afr. 1985, North-Holland Math. Stud. 126, 43-89 (1986).
[For the entire collection see Zbl 0588.00017.]
This is a review of the results known on the Jacobson radical of a semigroup ring R[S] of a commutative semigroup S over an associative ring R. The authors present with proofs all the steps leading to the final result describing J(R[S]) for rings R satisfying the condition \(J_ 1(R)=J_ n(R)\) for any \(n\geq 1\). Here \(J_ n(R)\) is defined through the radical of the polynomial ring by the equality \(J_ n(R)=J(R[x_ 1,...,x_ n])\cap R\). The description involves some ideals of R[S] related to the least separative and p-separative congruences on S as well as a complex component coming from the interrelations of the periodic subsemigroup of S with the archimedean decomposition of S. The final results are mainly taken from a forthcoming paper of E. Jespers [The Jacobson radical of semigroup rings of commutative semigroups] and a paper of the reviewer and P. Wauters [Math. Proc. Camb. Philos. Soc. 99, 435-445 (1986; see the following review)]. As an application, a discussion of when J(R[S]) is an S-homogeneous ideal of R[S] is given and semilocal rings R[S] are characterized. This covers the work done in two forthcoming papers of E. Jespers [When is the Jacobson radical of a semigroup ring of a commutative semigroup homogeneous] and P. Wauters, E. Jespers [When is a semigroup ring of a commutative semigroup local or semilocal].
Reviewer: J.Okniński

MSC:

20M25 Semigroup rings, multiplicative semigroups of rings
16Nxx Radicals and radical properties of associative rings
20M14 Commutative semigroups