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 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 J₂, ₁ domain (as defined by Lorenzini) is an elementary divisor domain.