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Weighted \(w\)-core inverses in rings. (English) Zbl 07729525

Summary: Let \(R\) be a unital \(\ast\)-ring. For any \(a,s,t,v,w\in R\) we define the weighted \(w\)-core inverse and the weighted dual \(s\)-core inverse, extending the \(w\)-core inverse and the dual \(s\)-core inverse, respectively. An element \(a\in R\) has a weighted \(w\)-core inverse with the weight \(v\) if there exists some \(x\in R\) such that \(awxvx=x\), \(xvawa=a\) and \((awx)^*=awx\). Dually, an element \(a\in R\) has a weighted dual \(s\)-core inverse with the weight \(t\) if there exists some \(y\in R\) such that \(ytysa=y\), \(asaty=a\) and \((ysa)^*=ysa\). Several characterizations of weighted \(w\)-core invertible and weighted dual \(s\)-core invertible elements are given when weights \(v\) and \(t\) are invertible Hermitian elements. Also, the relations among the weighted \(w\)-core inverse, the weighted dual \(s\)-core inverse, the \(e\)-core inverse, the dual \(f\)-core inverse, the weighted Moore-Penrose inverse and the \((v,w)\)-\((b,c)\)-inverse are considered.

MSC:

15A09 Theory of matrix inversion and generalized inverses
06A06 Partial orders, general
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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