COMPUTING THE EXTENDED CENTROID OF ABELIAN-GROUP GRADED RINGS

Let R be a prime ring and G a free multiplicative monoid of rank n. By R * G we denote a semigroup crossed product of R over G (see [9] for the definition). If n = 1, then R * G is a skew polynomial ring R[X, sigma], where sigma is an automorphism of R. In this case J. Matczuk shows in [5] that the extended centroid of R[X, sigma] is equal to ZZ-1, where Z is the center of Q[X, sigma] and Q is the left Martindale ring of quotients of R. In the case n greater-than-or-equal-to 2, in particular G is non-abelian, D.S. Passman in [8, 9] shows that R * G is symmetrically closed whenever R is symmetrically closed and cohesive. Hence the extended centroid of such rings is just the center of R * G. In case G is a group, there are additional examples in [8, 9] where the symmetric Martindale ring of quotients, and thus the extended centroid, is computed. However, in most of these examples the group G is non-abelian. In this paper we are interested in the case where G is an abelian group. We compute the extended centroid of a prime ring graded by an abelian group. As a corollary we obtain Matczuk's result.