A unit-picker is a map \({\mathcal {G}}\) that associates to every ring R a well-defined set \({\mathcal {G}}(R)\) of central units in R which contains \(1_R\) 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 \({\mathcal {G}}\) be a unit-picker. An element q of a ring R is \({\mathcal {G}}\) -idempotent, a special kind of the strongly regular element, if \(q^{2}=uq\) 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 \({\mathcal {G}}\) -projection if \(q^2=uq=uq^{*}\) 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 \({\mathcal {G}}\) -projection element.
