Aequat. Math. 92 (2018), 375–383
Springer International Publishing AG,
part of Springer Nature 2017
published online December 27, 2017
Invariance under outer inverses
R. E. Hartwig and P. Patr
Abstract. We shall use the minus partial order combined with Pierce’s decomposition to
derive the class of outer inverses for idempotents, units and group invertible elements. Sub-
sequently we show, for matrices over a ﬁeld F, that the triplet B
AC is invariant under all
choices of outer inverses of A if and only if B =0orC =0.
Mathematics Subject Classiﬁcation. 15A09, 16E50.
Keywords. Invariance, Outer inverses, Regularity, Pierce decomposition.
Let R be a ring with 1.
An element a is called regular if aa
a = a for some inner or 1-inverse
. The condition for regularity is a linear condition, and the set of all inner
inverses is given by
} = a
a)R + R(1 − aa
An outer or 2-inverse ˆa of an element a is such that ˆaaˆa =ˆa.Itisa
quadratic condition in ˆa. It is clear that aˆaa will always be regular.
A 1–2 or reﬂexive inverse of a is denoted by a
a = a and a
The set of all outer inverses of an element a will be denoted by T
and the set of all idempotents will be denoted by E. It is clear that a regular
element a admits a
as an outer inverse.
Given the quadratic nature of the outer inverse condition, the characteri-
zation of T
remains clouded in general. We shall characterize T
Partially supported by FCT—‘Funda¸c˜aoparaaCiˆencia e a Tecnologia’, within the project