Matrix invertible extensions over commutative rings. Part III: Hermite rings

We reobtain and often refine prior criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for a Hermite ring to be an EDR. We mention three criteria: (1) a Hermite ring R is an EDR iff for all pairs (a, c) R², the product homomorphism U (R/Rac) U (R/Rc (1-a) ) U (R/Rc) between groups of units is surjective; (2) a reduced Hermite ring is an EDR iff it is a pre-Schreier ring and for each a R, every zero determinant unimodular 2 2 matrix with entries in R/Ra lifts to a zero determinant matrix with entries in R; (3) a B\'ezout domain R is an EDD iff for all triples (a, b, c) R³ there exists a unimodular pair (e, f) R² such that (a, e) and (be+af, 1-a-bc) are unimodular pairs. We use these criteria to show that each B\'ezout ring R that is an (SU) ₂ ring (as introduced by Lorenzini) such that for each nonzero a R there exists no nontrivial self-dual projective R/Ra-module of rank 1 generated by 2 elements (e. g. , all its elements are squares), is an EDR.