A class of outer generalized inverses. (English) Zbl 1254.15005

Drazin, Michael P.

Linear Algebra Appl. 436, No. 7, 1909-1923 (2012).

The notion of the Drazin generalized inverse in associative rings and semi-groups, introduced by the author of the present article [Am. Math. Mon. 65, 506–514 (1958; Zbl 0083.02901)] is as follows: Let \(S\) be a semigroup. An element \(b \in S\), is called the Drazin inverse of \(a\) if \(ab=ba,b=bab\) and \(a^{k+1}b=a^k\) for some nonnegative integer \(k\). Such a \(b\) is unique and is denoted by \(a^D\). Let us also recall the more widely known generalized inverse. This time, let \(S\) be a multiplicative semigroup with a given involution \(*\). For \(a \in S\) it is the Moore-Penrose inverse \(a^{\dagger}\) which is defined as the (unique) element \(b \in S\) that satisfies the equations: \(aba=a, bab=b; (ab)^*=ab\) and \((ba)^*=ba\). Any \(b \in S\) that satisfies the equation \(bab=b\) is called an outer inverse of \(a \in S\). Clearly, the two generalized inverses given earlier are specific outer inverses. In the paper under review, the author outlines a unified theory emcompassing a large class of uniquely-defined outer generalized inverses, especially including the Moore-Penrose inverse and the Drazin inverse. This theory is developed by exploiting the similarity and differences between these two classes of outer generalized inverses.

Reviewer: K. C. Sivakumar (Chennai)

MSC:

15A09	Theory of matrix inversion and generalized inverses
06A11	Algebraic aspects of posets
16P99	Chain conditions, growth conditions, and other forms of finiteness for associative rings and algebras
20M99	Semigroups

Keywords:

Bott-Duffin inverse; computation of generalized inverses; exchange ring; extremal properties; Mitsch partial order; Moore-Penrose generalized inverse; outer generalized inverses; potent ring; pseudo-inverse; semigroup; stable range one; strong \(\pi \)-regularity; strongly clean ring; suitable ring; weighted inverse; \(W\)-weighted pseudo-inverse; Drazin generalized inverse

Citations:

Zbl 0083.02901

Software:

freegb.lib; Letterplace; SINGULAR

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