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.
Reviewer: David F. Anderson (Knoxville)
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 |
Keywords:
trivial extension; pullback; coherence; finite conductor ring; \(v\)-coherent ring; quasi-coherent ring; \(n\)-coherence; G-GCD rings; \((n,d)\)-ringsReferences:
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