Inner inverses and inner annihilators in rings

For any ring element α∈Rα∈R, we study the group of inner annihilators IAnn(α)={p∈R:αpα=0}IAnn(α)={p∈R:αpα=0} and the set I(α)I(α) of inner inverses of α  . For any Jacobson pair α=1−abα=1−ab and β=1−baβ=1−ba, the groups A=IAnn(α)A=IAnn(α) and B=IAnn(β)B=IAnn(β) are shown to be equipotent, and A⊕CA⊕C is shown to be group isomorphic to B⊕CB⊕C where C=Annℓ(α)⊕Annr(α)C=Annℓ(α)⊕Annr(α). In the case where α   is (von Neumann) regular, we show further that A≅BA≅B as groups. For any Jacobson pair {α,β}{α,β}, a “new Jacobson map” Φ:I(α)→I(β)Φ:I(α)→I(β) is constructed that is a semigroup homomorphism with respect to the von Neumann product, and preserves units, reflexive inverses and commuting inner inverses. In particular, for any abelian ring R, Φ   is a semigroup isomorphism between I(α)I(α) and I(β)I(β). As a byproduct of our methods, we also show that a ring R satisfies internal cancellation iff every Jacobson pair of regular elements are equivalent over R. In particular, the latter property holds for many rings, including semilocal rings, unit-regular rings, strongly π-regular rings, and finite von Neumann algebras.