Matrix invertible extensions over commutative rings. Part II: Determinant liftability

Abstract

A unimodular $2\times 2$ matrix A with entries in a commutative ring R is called weakly determinant liftable if there exists a matrix B congruent to A modulo $R\det \left(A\right)$ and $\det \left(B\right)=0$; if we can choose B to be unimodular, then A is called determinant liftable. If A is extendable to an invertible $3\times 3$ matrix $A^{+}$, then A is weakly determinant liftable. If A is simply extendable (i.e., we can choose $A^{+}$ such that its $(3,3)$ entry is 0), then A is determinant liftable. We present necessary and/or sufficient criteria for A to be (weakly) determinant liftable and we use them to show that if R is a ${\Pi }_2$ ring in the sense of Part I (resp. is a pre-Schreier domain), then A is simply extendable (resp. extendable) iff it is determinant liftable (resp. weakly determinant liftable). As an application we show that each $J_{2,1}$ domain (as defined by Lorenzini) is an elementary divisor domain.