The determinal rank idempotents of a matrix Original Research Article

Let MR be the category of finite matrices over a commutative ring R with 1. For A set membership, variant MR of determinantal rank r let imager(A) be the ideal of R generated by the r × r minors of A. There exists a unique list (e1, …, et) of pairwise orthogonal idempotents of R that sum to 1 such that, if ri = rank(eiA), then rank A = r1 > r2 > … > rt, ei is the identity element of imageri(eiA) for 1 ≤ i < t, and either etA = 0 or imagert(etA) does not possess an identity element. Characterizations are given for various generalized inverses of A = e1A + … + etA. In particular, A is regular iff etA = 0.