Strongly prime and $$*$$ ∗ -prime crossed products

Abstract

Let \(R*G\) be the crossed product of the group G over the ring R. In this paper, we obtain conditions on the group G and the normal subgroup H of G which ensure that \(R*G\) is right strongly prime if and only if \(R*H\) is right strongly prime under the assumption that R is prime. Also we show that if R is prime, G is a finite group and \(R * G\) is SP, then R is SP. It is proved that if R is SP, G is FC-solvable group with \(L(G) = \langle 1\rangle \), then \(R*G\) is SP. This extends the results of Sharma and Srivastava (NZ J Math 23:93–98, 1994) from group rings to crossed products. Parallel to the notion of \(*\)-prime group rings introduced in Joshi et al. (Commun Algebra 35:3673–3682, 2007), we define \(*\)-prime crossed products and show that if R is a prime ring and H is a normal subgroup of G with \(D_G(H) = \langle 1 \rangle \), then \(R * H\) is \(*\)-prime implies \(R * G\) is \(*\)-prime. Also we prove that if \(R* G\) is \(*\)-prime, H is a subgroup of G with \(\vert G : H \vert < \infty \) and \(\overline{x}^{-1}\alpha \overline{x}= \alpha \) for all \(x \in G\) and for all \(\alpha \in R*H\), then \(R*H\) is \(*\)-prime.