Rank factorization and bordering of regular matrices over commutative rings
Let \(R\) be a commutative ring and \(A\) be an \(n \times m\) matrix over \(R\). An elementary extension of \(A\) is a matrix denoted by \(A(\alpha,\beta)\) for some (non-negative) integers \(\alpha\) and \(\beta\), which is of size \((m+\alpha+\beta)\times (n+\beta)\) with \(A\) in the left corner, the \(\beta \times \beta\) identity matrix along the diagonal and with 0 in the remaining places. The author shows that if the maximal spectrum of \(R\) is a Noetherian space of dimension \(d\) and \(A\) is a regular \(m \times n\) matrix of rank \(r\) whose associated module \(Ker(A)\) is stably free, then \(A(\alpha,\beta)\) has a rank factorization with \(\alpha= \max(d+1+r-n,0)\) and \(\beta= \max(d+1-r,0)\). The author also discusses the existence of \(f\) in \(R\) such that borderings and rank factorizations of regular matrices over \(R\) are true allowing \(f\) as denominator. The paper under review has its origin in the work of \textit{K. Manjunatha Prasad} and \textit{K. P. S. Bhaskara Rao} [Linear Algebra Appl. 234, 245-259 (1996; Zbl 0841.15009)].
scientific article; zbMATH DE number 1419722