Generalized inverses over integral domains
In an earlier paper [Biol. 49, 179--189 (1983; Zbl 0505.15002)] the second author showed that a matrix of rank \(r\) over an integral domain has a generalized inverse if and only if a linear combination of all minors of the matrix is one. He also gave a procedure for constructing a generalized inverse from a linear combination of minors. This paper shows that any reflective generalized inverse can be obtained by this procedure. In a principal ideal domain the procedure gives all generalized inverses. Finally, a rank \(r\) matrix over an integral domain has a Moore-Penrose inverse if and only if the sum of squares of all \(r\times r\) minors is an invertible element of the integral domain. Many papers on generalized inverses over various types of rings have been published during the last twenty years. This paper has only four references.
scientific article; zbMATH DE number 4171110