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Classes of \(D+M\) rings defined by homological conditions. (English) Zbl 0854.13013

One says that a commutative ring \(R\) is an \((n, d)\)-ring if every \(R\)-module having a finite \(n\)-presentation has projective dimension at most \(d\). Here, an \(R\)-module \(M\) has a finite \(n\)-presentation if there is an exact sequence \[ F_n\to F_{n-1}\to \cdots\to F_1\to F_0\to M\to 0 \] in which each \(F_i\) is a finitely generated free \(R\)-module. In general, a commutative ring \(R\) is called a weak \((n, d)\)-ring if every finitely \(n\)-presented cyclic \(R\)-module has projective dimension at most \(d\). In this article, the authors prove:
Theorem 1. Let \(T\) be an integral domain of the form \(K+M\), where \(K\) is a field and \(M\) is a maximal ideal of \(T\). Let \(D\) be a proper subring of \(K\), and let \(k=Q (D)\) be the quotient field of \(D\). Put \(R= D+ M\). Suppose that \(T\) is a \((1, 1)\)-domain. Then the following are equivalent:
(i) \(R\) is a \((1, 2)\)-domain;
(ii) \(D\) is a \((1, 2)\)-domain and \(k= K\).
Theorem 2. With notation as in theorem 1, suppose that \(T\) is a valuation ring and either: (a) \([K: k]= \infty\), or (b) \([K: k]< \infty\) and \(M= M^2\), or (c) \(K=k\). Then
(i) \(R\) is a \((2, 1)\)-domain if and only if \(D\) is a \((2, 1)\)-domain.
(ii) \(R\) is a weak \((2, 1)\)-domain if and only if \(D\) is a weak \((2, 1)\)-domain.

MSC:

13G05 Integral domains
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13D05 Homological dimension and commutative rings
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References:

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