The determinal rank idempotents of a matrix
Let \({\mathbf M}_R\) be the category of finite matrices over a commutative ring \(R\) with 1. For \(A\in {\mathbf M}_R\) of determinantal rank \(r\) let \({\mathcal I}_r (A)\) be the ideal of \(R\) generated by the \(r\times r\) minors of \(A\). There exists a unique list \((e_1, \dots, e_t)\) of pairwise orthogonal idempotents of \(R\) that sum to 1 such that, if \(r_i= \text{rank} (e_i A)\), then \(\text{rank } A= r_1> r_2> \dots> r_t\), \(e_i\) is the identity element of \({\mathcal I}_{r_i} (e_i A)\) for \(1\leq i<t\), and either \(e_t A=0\) or \({\mathcal I}_{r_t} (e_t A)\) does not possess an identity element. Characterizations are given for various generalized inverses of \(A= e_1 A+ \dots+ e_t A\). In particular, \(A\) is regular if and only if \(e_t A=0\).
scientific article; zbMATH DE number 885060