Closed and prime ideals in partial skew group rings of abelian groups

Abstract

In this paper we study partial actions of abelian groups on a ring R having an enveloping action. Among other results we prove that if α is such a partial action and (T, β) its globalization, then there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R *α G and closed (respectively, T-disjoint prime) ideals of T *β G. We also prove that there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R *α G and closed (respectively, Q-disjoint prime) ideals of Q*α G, where Q is the left α-quotient ring of R. Finally, we use these results to study strongly prime ideals and nonsingular prime ideals of R *α G. © 2011 World Scientific Publishing Company.