Unit-regularity of elements in rings

Abstract

An element [Formula: see text] in a unital ring [Formula: see text] is said to have an inverse complement [Formula: see text] if [Formula: see text] is a unit of [Formula: see text] and [Formula: see text]. Unit-regular elements are studied from the viewpoint of the existence of inverse complements. As a source of unit-regular elements, we prove that if [Formula: see text] is a completely reducible submodule of [Formula: see text], then every element of [Formula: see text] is unit-regular if and only if any nonzero submodule of [Formula: see text] is not square zero. This generalizes some results due to Stopar in 2020. Finally, extending the case of real or complex matrices to the context of rings, we characterize the outer and reflexive inverses of a given unit-regular element depending only on its inverse complement.