Products of elementary and idempotent matrices over integral domains

A ring R such that invertible matrices over R   are products of elementary matrices, is called (after Cohn) generalized Euclidean. Extending results proved by Ruitenburg for Bézout domains, characterizations of generalized Euclidean commutative domains are obtained, that relate them with the property that singular matrices are products of idempotent matrices. This latter property is investigated, focusing on 2×22×2 matrices, which is not restrictive in the context of Bézout domains. It is proved that domains R  , that satisfy a suitable property of ideals called (princ), are necessarily Bézout domains if 2×22×2 singular matrices over R are products of idempotent matrices. The class of rings satisfying (princ) includes factorial and projective-free domains. The connection with the existence of a weak Euclidean algorithm, a notion introduced by O'Meara for Dedekind domains, is also investigated.