关于$\ast$ -环的广义逆和偏序
On some generalized inverses and partial orders in $\ast$-rings
摘要：
设$\mathcal{R}$是一个具有对合的酉环.本文将1 MP-逆和MP 1-逆的概念从$M_{m,n}(\mathbb{C)}$（$\mathbb{C}$上所有$m\times n $矩阵的集合）推广到$\mathcal{R}$中所有Moore-Penrose可逆元的集合$\mathcal{R}% ^{\dagger}$。研究了$\mathcal{R}^{\dagger}$上由1 MP-逆和MP 1-逆诱导的偏序.我们还扩展到设置的Rickart $\ast $ -环的另一个偏序的概念，称为加号顺序，这是最近推出的集合上的所有有界线性算子之间的希尔伯特空间。研究了这些关系的性质，推广了已有的结果。
英文摘要：
Let $\mathcal{R}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(\mathbb{C)}$, the set of all $m\times n $ matrices over $\mathbb{C}$, to the set $\mathcal{R}% ^{\dagger}$ of all Moore-Penrose invertible elements in $\mathcal{R}$. We study partial orders on $\mathcal{R}^{\dagger}$ that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart $\ast $-rings the concept of another partial order, called the plus order, which has been recently introduced on the set of all bounded linear operators between Hilbert spaces. Properties of these relations are investigated and some known results are thus generalized.
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