Partial crossed products and fully weakly prime rings. (English) Zbl 1383.16026

Cortes, Wagner; Soares, Marlon

Rocky Mt. J. Math. 46, No. 4, 1107-1139 (2016).

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 \[ \text{Nil}_{*}(R*_{\alpha}^{w}G)=\text{Nil}(R)*_{\alpha}^{w}G=\text{Nil}_{\alpha}(R)*_{\alpha}^{w}G=\text{Nil}_{*}(R)*_{\alpha}^{w}G. \] 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 \(K\subseteq I\).
Necessary and sufficient conditions for a partial crossed product to be fully weakly prime are also studied.

Reviewer: Wen-Fong Ke (Tainan)

MSC:

16S35	Twisted and skew group rings, crossed products
16N60	Prime and semiprime associative rings
16W22	Actions of groups and semigroups; invariant theory (associative rings and algebras)

Keywords:

partial crossed product; fully weakly prime rings; prime radical

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