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Finite homomorphic images of Bezout duo-domains. (English) Zbl 1321.16024

Summary: 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.

MSC:

16U30 Divisibility, noncommutative UFDs
16U80 Generalizations of commutativity (associative rings and algebras)
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
19B10 Stable range conditions
16E20 Grothendieck groups, \(K\)-theory, etc.
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