Finite homomorphic images of Bezout duo-domains

Abstract

It is proved that for a quasi-duo Bezout ring of stable range 1 the duo-ring condition is equivalent to being an elementary divisor ring. As an application of this result a couple of useful properties are obtained for finite homomorphic images of Bezout duo-domains: they are coherent morphic rings, all injective modules over them are flat, their weak global dimension is either 0 or infinity. Moreover, we introduce the notion of square-free element in noncommutative case and it is shown that they are adequate elements of Bezout duo-domains. In addition, we are going to prove that these elements are elements of almost stable range 1, as well as necessary and sufficient conditions for being square-free element are found in terms of regularity, Jacobson semisimplicity, and boundness of weak global dimension of finite homomorphic images of Bezout duo-domains.

Elementary matrix reduction over Bézout domains

“…In view of [11, Theorem 17], a or 1 − a is sqaure-free. It follows by [11,Proposition 16] that a or 1 − a is adequate in R. Therefore we complete the proof, by Theorem 3.11. ✷ A ring R is called homomorphically semiprimitive if every ring homomorphic image (including R) of R has zero Jacobson radical.…”

“…By hypothesis, J( R aR ) = 0 or J( R (1−a)R ) = 0. In view of [11,Theorem 17], a or 1 − a is sqaure-free. It follows by [11,Proposition 16] that a or 1 − a is adequate in R. Therefore we complete the proof, by Theorem 3.11.…”