# zbMATH — the first resource for mathematics

Radicals and semigroup rings. (Russian) Zbl 0676.16007
A 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 $$xy=\min (x,y))$$.
Reviewer: J.S.Ponizovskij

##### MSC:
 16Nxx Radicals and radical properties of associative rings 20M25 Semigroup rings, multiplicative semigroups of rings 20M10 General structure theory for semigroups