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

• Published in: Linear algebra and its applications Vol. 725; pp. 172 - 197
• Main Authors: Călugăreanu, Grigore, Pop, Horia F., Vasiu, Adrian
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.11.2025
• Subjects: Matrix; Projective module; Ring; Unimodular; secondary; Projective module; Matrix; Unimodular; Ring; primary
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2025.07.008

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 Rdet⁡(A) and det⁡(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 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 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.