Partial crossed products and fully weakly prime rings (Q329912)

Given a twisted partial action \(\alpha\) of a group \(G\) on a ring \(R\), the partial crossed product \(R*_{\alpha}^{w}G\) defines an associative ring. When \(R\) is fully weakly prime ring, it is shown that NEWLINE\[NEWLINE \text{Nil}_{*}(R*_{\alpha}^{w}G)=\text{Nil}(R)*_{\alpha}^{w}G=\text{Nil}_{\alpha}(R)*_{\alpha}^{w}G=\text{Nil}_{*}(R)*_{\alpha}^{w}G. NEWLINE\]NEWLINE Here, \(R\) is said to be fully weakly prime when every proper ideal \(I\) of \(R\) satisfies that for any ideals \(J\) and \(K\) of \(R\) with \(0\not=JK\subseteq I\), either \(J\subseteq I\) or \(K\subseteq I\).NEWLINENEWLINENecessary and sufficient conditions for a partial crossed product to be fully weakly prime are also studied.