Radicals and semigroup rings.

*(Russian)*Zbl 0676.16007A ring means an associative ring, a radical means a ring radical in the sense of Kurosh-Amitsur. A denotes the class of all semigroups S such that \(r(R[S])=r(R)[S]\) for all rings R and for all radicals r. The aim of the paper is to investigate semigroups from A. Let \(\rho\) be a radical such that, for any \(\rho\)-radical ring R, any nonzero homomorphic image of R contains a nonzero ideal isomorphic to a factor-ring of the ring \({\mathbb{Z}}\) of integers. Let S be a semigroup, and let K be a class of rings; then K is S-closed iff \(R\in K\) implies R[S]\(\in K\) for any ring R.

The main theorem states that the following conditions on a semigroup S are equivalent: (i) \(S\in A\); (ii) for any radical r, the class of all r- radical rings is S-closed; (iii) S is a semilattice such that the semigroup ring \({\mathbb{Z}}[S]\) is \(\rho\)-radical. The author shows that the class A is closed under homomorphisms and subsemigroups, that any semilattice satisfying the minimum condition belongs to A, that a nonidentical chain belongs to A if and only if it does not contain a subsemigroup isomorphic to a semigroup which the author denotes by \(Q_ 2\) \((Q_ 2\) consists of all rational numbers \(x=-m\cdot 2^{-n}\) where m, n are nonnegative integers, \(m<2^ n\), the multiplication in \(Q_ 2\) being defined by the rule \(xy=\min (x,y))\).

The main theorem states that the following conditions on a semigroup S are equivalent: (i) \(S\in A\); (ii) for any radical r, the class of all r- radical rings is S-closed; (iii) S is a semilattice such that the semigroup ring \({\mathbb{Z}}[S]\) is \(\rho\)-radical. The author shows that the class A is closed under homomorphisms and subsemigroups, that any semilattice satisfying the minimum condition belongs to A, that a nonidentical chain belongs to A if and only if it does not contain a subsemigroup isomorphic to a semigroup which the author denotes by \(Q_ 2\) \((Q_ 2\) consists of all rational numbers \(x=-m\cdot 2^{-n}\) where m, n are nonnegative integers, \(m<2^ n\), the multiplication in \(Q_ 2\) being defined by the rule \(xy=\min (x,y))\).

Reviewer: J.S.Ponizovskij