The Brown Mccoy radical of semigroup rings of commutative cancellative semigroups

Abstract

We give a complete description of the Brown–McCoy radical of a semigroup ring R[S], where R is an arbitrary associative ring and S is a commutative cancellative semigroup; in particular we obtain the answer to a question of E. Puczyłowski stated in [11]Throughout this note all rings R are associative with unity 1; all semigroups S are commutative and cancellative with unity. Note that the condition that R and S have a unity can be dropped (cf. [8]). The quotient group of S is denoted by Q(S). We say that S is torsion free (resp. has torsion free rank n) if Q(S) is torsion free (resp. has torsion free rank n). The Brown–McCoy radical (i.e. the upper radical determined by the class of all simple rings with unity) of a ring R is denoted by u(R). We refer to [2] for further detail on radicals and in particular on the Brown–McCoy radical.First we state some well-known results and a preliminary lemma. Let R and T be rings with the same unity such that R ⊂ T. Then T is said to be a normalizing extension of R if T = Rx1+…+Rxn for certain elements x1, …, xn of T and Rxi = xiR for all i such that 1 ∨i∨n. If all xi are central in T, then we say that T is a central normalizing extension of R.