When is the Jacobson radical of a semigroup ring of a commutative semigroup homogeneous?
Let S be a commutative semigroup and let R be an associative ring such that \(R\cap J(R[x])=R\cap J(R[x_ 1,...,x_ n])\) for any \(n>1\) where J denotes the Jacobson radical. Based on the author's description of J(R[S]) [ibid. 109, 266-280 (1987; Zbl 0617.20040)], necessary and sufficient conditions for J(R[S]) to be S-homogeneous are determined. In particular, it is shown that in this case \(J(R[S])=(R\cap J(R[S]))[S]+J(R)[P]\) where P denotes the largest ideal of S consisting of periodic elements. While in general the result obtained does not provide a complete intrinsic characterization of the homogeneity of J(R[S]) in terms of S, such a result is obtained for some special classes of semigroups and the difficulties when dealing with the general case are illustrated.
scientific article; zbMATH DE number 4004487
scientific article