Abstract

We extend characterizations of inverses along an element to (b, c)-inverses. It is proved that if an element a in a semigroup S is both (b, c)-invertible and (c, b)-invertible for some $b,c\in S,$ then $\mathit{abac},\mathit{acab},\mathit{caba},\mathit{baca}$ are group invertible, and $\mathit{bac}{(\mathit{abac})}^{♯}=ba{(\mathit{caba})}^{♯}c=b{(\mathit{acab})}^{♯}ac={(\mathit{baca})}^{♯}bac$ is the (b, c)-inverse of a, $\mathit{cab}{(\mathit{acab})}^{♯}=ca{(\mathit{baca})}^{♯}b=c{(\mathit{abac})}^{♯}ab={(\mathit{caba})}^{♯}cab$ is the (c, b)-inverse of a. Moreover, we establish several criteria for the existence of (b, c)-inverses and (c, b)-inverses by means of units in a ring. As applications, some new representations of weighted Moore-Penrose inverses and core-EP inverses of complex matrices are given.

摘要

我们将逆元的特征扩展到 ( b ,  c ) 逆元。证明如果半群S 中的元素a既是 ( b ,  c )-可逆的又是 ( c ,  b )-可逆的$乙,C\in 秒,$ 然后 $\mathit{算盘},\mathit{驾驶室},\mathit{卡巴},\mathit{巴卡}$ 是群可逆的，并且 $\mathit{巴克}{(\mathit{算盘})}^{♯}=乙\mathrm{一种}{(\mathit{卡巴})}^{♯}C=乙{(\mathit{驾驶室})}^{♯}\mathrm{一种}C={(\mathit{巴卡})}^{♯}乙\mathrm{一种}C$是（b，  c ^）的-逆一个，$\mathit{出租车}{(\mathit{驾驶室})}^{♯}=C\mathrm{一种}{(\mathit{巴卡})}^{♯}乙=C{(\mathit{算盘})}^{♯}\mathrm{一种}乙={(\mathit{卡巴})}^{♯}C\mathrm{一种}乙$是（C ^，  b）的-逆一个。此外，我们通过环中的单元为 ( b ,  c )-逆和 ( c ,  b )-逆的存在建立了几个标准。作为应用，给出了复矩阵的加权 Moore-Penrose 逆和 core-EP 逆的一些新表示。