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\begin{document}$$R*G$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$R*G$$\end{document} is right strongly prime if and only if R∗H\documentclass[12pt]{minimal}
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\begin{document}$$R*H$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$R * G$$\end{document} is SP, then R is SP. It is proved that if R is SP, G is FC-solvable group with L(G)=⟨1⟩\documentclass[12pt]{minimal}
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\begin{document}$$L(G) = \langle 1\rangle $$\end{document}, then R∗G\documentclass[12pt]{minimal}
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\begin{document}$$R*G$$\end{document} 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 ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-prime group rings introduced in Joshi et al. (Commun Algebra 35:3673–3682, 2007), we define ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-prime crossed products and show that if R is a prime ring and H is a normal subgroup of G with DG(H)=⟨1⟩\documentclass[12pt]{minimal}
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\begin{document}$$D_G(H) = \langle 1 \rangle $$\end{document}, then R∗H\documentclass[12pt]{minimal}
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\begin{document}$$R * H$$\end{document} is ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-prime implies R∗G\documentclass[12pt]{minimal}
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\begin{document}$$R * G$$\end{document} is ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-prime. Also we prove that if R∗G\documentclass[12pt]{minimal}
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\begin{document}$$R* G$$\end{document} is ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-prime, H is a subgroup of G with |G:H|<∞\documentclass[12pt]{minimal}
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\begin{document}$$\vert G : H \vert < \infty $$\end{document} and x¯-1αx¯=α\documentclass[12pt]{minimal}
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\begin{document}$$\overline{x}^{-1}\alpha \overline{x}= \alpha $$\end{document} for all x∈G\documentclass[12pt]{minimal}
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\begin{document}$$x \in G$$\end{document} and for all α∈R∗H\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in R*H$$\end{document}, then R∗H\documentclass[12pt]{minimal}
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\begin{document}$$R*H$$\end{document} is ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-prime.
