A generalization of the generalized Drazin inverse in rings

The notions of (quasi-) polarity and (generalized) Drazin invertibility of an element a in a ring R, defined upon the set of (quasi-) nilpotent elements, are extended by using the set ∇(R) = {a ∈ R : 1 − au is unit in R for all unit u ∈ R with ua = au}. This set is adapted from the set ∆(R), the largest Jacobson radical subring of R closed by multiplication by units. Uniqueness of the spectral idempotent usually associated to (quasi-) polar decompositions remains, and can be defined by minimal properties. This leads to the definition of a unique Drazin-type inverse for a wider collection of decompositions, encompassing the quasi-polar decompositions. © World Scientific Publishing Company.