The determinal rank idempotents of a matrix

Abstract

Let M R be the category of finite matrices over a commutative ring R with 1. For A ∈ M R of determinantal rank r let I r ( A) be the ideal of R generated by the r × r minors of A. There exists a unique list ( e 1, …, e t ) of pairwise orthogonal idempotents of R that sum to 1 such that, if r i = rank( e i A), then rank A = r 1 > r 2 > … > r t , e i is the identity element of I r i ( e i A) for 1 ≤ i < t, and either e t A = 0 or I r t ( e t A) does not possess an identity element. Characterizations are given for various generalized inverses of A = e 1 A + … + e t A. In particular, A is regular iff e t A = 0.