Geodesic
Weighted $w$-core inverses in rings
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 581-602
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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.
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.
DOI : 10.21136/CMJ.2022.0134-22
Classification : 06A06, 15A09, 16W10
Keywords: inverse along an element; $\{e, 1, 3\}$-inverse; ${\{f, 1, 4}\}$-inverse; weighted Moore-Penrose inverse; $(v, w)$-$(b, c)$-inverse; $w$-core inverse; dual $v$-core inverse 
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Wu, Liyun; Zhu, Huihui. Weighted $w$-core inverses in rings. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 581-602. doi: 10.21136/CMJ.2022.0134-22

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