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

A unimodular 2× 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(A) and (B)=0; if we can choose B to be unimodular, then A is called determinant liftable. If A is extendable to an invertible 3× 3 matrix A⁺, then A is weakly determinant liftable. If A is simple 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 Π₂ 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 J2,1 domain (as defined by Lorenzini) is an elementary divisor domain.