A generalization of the generalized Drazin inverse in rings

The notions of (quasi-) polarity and (generalized) Drazin invertibility of an element [Formula: see text] in a ring [Formula: see text], defined upon the set of (quasi-) nilpotent elements, are extended by using the set [Formula: see text] This set is adapted from the set [Formula: see text], the largest Jacobson radical subring of [Formula: see text] 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.