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\'{e}zout 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\'{e}zout 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}.