Characterization of ∗-(strongly) regular rings in terms of G-projections

A unit-picker is a map G that associates to every ring R a well-defined set G(R) of central units in R which contains 1R and is invariant under isomorphisms of rings and closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let G be a unit-picker. An element q of a ring R is G-idempotent, a special kind of the strongly regular element, if q2=uq for some unit picker u of R, or equivalently, q=ue, where u is a unit picker and e is an idempotent of R. In a ring R with involution ∗, projections are self-adjoint idempotents. As a natural generalization of projections, an element q of a ring R is called a G-projection if q2=uq=uq∗ for some self-adjoint unit-picker u of a ∗-ring R, or equivalently, q=up, where p is a projection. We characterize ∗-(strongly) regular rings in terms of the G-projection element. © The Indian National Science Academy 2024.