### The Brown Mccoy radical of semigroup rings of commutative cancellative semigroups

E. Jespers, J. Krempa, P. Wauters
1985 Glasgow Mathematical Journal
1. 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. PuczyTowski stated in . 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. ). The quotient group of S is denoted by
more » ... 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 %l(R). We refer to  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 c T . Then T is said to be a normalizing extension of R if T = Rx^ +... + RXn for certain elements x u ..., x n of T and i?x f = x ; l? for all i such that 1 ^ i =£ n. If all x t are central in T, then we say that T is a central normalizing extension of R. PROPOSITION 1.1. Let R and T be rings such that T is a normalizing extension of R. Then Proof, cf.  or . PROPOSITION 1.2. (1) Let G be a finite abelian group of order n and let R be a G-graded ring. If a = £ « g e<U(R), then na g e<U(R) for all g e G. gsG (2) // S is a torsion free commutative semigroup and if R is an S-graded ring, then <U(R) is homogeneous, i.e. if £ r s e%(i?), then r s e<&(R) for all s. s