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Trivial extensions defined by coherent-like conditions. (English) Zbl 1068.13002

In this article, the authors investigate coherent-like and finiteness properties of the trivial ring extension \(R\) of a ring \(A\) by an \(A\)-module \(E\) (all rings are commutative with identity). Their results recover previous results from the literature and generate several interesting new families of examples. They generalize the concept of \(v\)-coherent domains to rings with zero-divisors. Some of the other rings (with zero-divisors) they investigate include coherent, quasi-coherent, finite conductor, and G-GCD rings. The authors also study the \((n,d)\)-rings of D. Costa and \(n\)-coherent and strongly \(n\)-coherent rings.

MSC:

13B02 Extension theory of commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13D05 Homological dimension and commutative rings
13E99 Chain conditions, finiteness conditions in commutative ring theory
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