2024/05/02 by Călugăreanu, Grigore, Pop, Horia F., Vasiu, Adrian
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.01234
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 R 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 Bézout 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 Bézout 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.