Characterizations of m-weak group inverses

m-弱群逆的刻画

To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.

为了刻画m-弱群逆,采用幂等元刻画、单边主理想刻画、可逆元刻画等代数方法.设a为带对合的幺环中的Drazin可逆元.首先,给出了元素a的m-弱群逆的存在性刻画,得到a为伪核可逆元和弱群可逆元的等价条件.其次,在弱proper*-环中,证明元素a为弱群可逆的当且仅当a的Drazin逆为au的弱群逆,u为一个与a的Drazin逆密切相关的特殊可逆元.然后,给出一个反例说明a的Drazin逆一般不是其他元素的伪核逆,由此表明了伪核逆与弱群逆的区别.最后,将关于弱群逆交换性的结论推广到m-弱群逆的情形,给出了新的关于广义逆交换性的条件.结果表明,伪核逆、弱群逆与m-弱群逆联系密切,但又存在显著区别.