A characterization of Prüfer $v$-multiplication domains in terms of linear equations

Abstract

In this paper, we characterize Prüfer $v$-multiplication domains as integral domains which have the property that the existence of a generalized solution of any system of linear equations is equivalent to a weak equality of the determinantal ideals of the coefficient matrix and the augmented matrix of the system. In fact, we obtain a more general result for commutative rings of weak global $\tau$-dimension (in the sense of Bueso, Van Ostaeyen and Verschoren) at most one, where $\tau$ is a half-centered hereditary torsion theory.