Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2020-08-25 , DOI: 10.1007/s41980-020-00449-8 Abdelkarim Boua , Mahmoud Mohammed El-Soufi , Ahmed Yunis Abdelwanis
Let \({\mathcal {R}}\) be a semiprime ring with center \(Z({\mathcal {R}})\) and with extended centroid C and let \(\sigma : {\mathcal {R}} \rightarrow {\mathcal {R}}\) be an automorphism. Assume that \(\tau : {\mathcal {R}} \rightarrow {\mathcal {R}} \) is an anti-homomorphism, such that the image of \(\tau \) has small centralizer. It is proved that the following are equivalent: (1) \(x^{\sigma }x^{\tau } = x^{\tau }x^{\sigma }\) for all \(x\in {\mathcal {R}};\) (2) \(x^{\sigma } + x^{\tau }\in Z({\mathcal {R}})\) for all \(x\in {\mathcal {R}};\) (3) \(x^{\sigma }x^{\tau }\in Z({\mathcal {R}})\) for all \(x\in {\mathcal {R}}.\) In this case, there exists an idempotent \(e \in C\), such that \((1-e){\mathcal {R}}\) is a commutative ring and the semiprime ring \(e{\mathcal {R}}\) is equipped with an involution \(\widetilde{\tau }\), which is induced canonically by \(\tau \). Note that one can easily obtained the main result in Lee (Commun Algebra 46(3):1060–1065, 2018) when \(\sigma =id_{{\mathcal {R}}}.\)
中文翻译:
$$ \ sigma $$σ-通勤和$$ \ sigma $$σ-集中反同态
令\({\ mathcal {R}} \)是中心为\(Z({\ mathcal {R}})\)且质心为C的半素环,并令\(\ sigma:{\ mathcal {R} } \ rightarrow {\ mathcal {R}} \)是自同构的。假设\(\ tau:{\ mathcal {R}} \ rightarrow {\ mathcal {R}} \)是反同态的,因此\(\ tau \)的图像集中度较小。证明以下各项是等效的:(1 )对于所有\ {x \ in {\中的\(x ^ {\ sigma} x ^ {\ tau} = x ^ {\ tau} x ^ {\ sigma} \)mathcal {R}}; \)(2 )对所有\ {x \ in {\ mathcal中的)\(x ^ {\ sigma} + x ^ {\ tau} \ in Z({\ mathcal {R}})\){R}}; \)(3)\(x ^ {\ sigma} x ^ {\ tau} \ in Z({\ mathcal {R}})\)中所有\(x \ in {\ mathcal {R}}。\\)在这种情况下,存在一个幂等\(e \ in C \),因此\((1-e){\ mathcal {R}} \)是一个交换环,而半素环\(e {\ mathcal {R}} \)配备了对合\(\ widetilde {\ tau} \),该对合由\(\ tau \)规范地诱发。请注意,当\(\ sigma = id _ {{\ mathcal {R}}}。\)时,可以很容易地在Lee中获得主要结果(Commun Algebra 46(3):1060-1065,2018)。