Costa, Douglas L.; Kabbaj, Salah-Eddine Classes of \(D+M\) rings defined by homological conditions. (English) Zbl 0854.13013 Commun. Algebra 24, No. 3, 891-906 (1996). 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. Reviewer: J.Nishimura (Sapporo) Cited in 6 Documents MSC: 13G05 Integral domains 13E15 Commutative rings and modules of finite generation or presentation; number of generators 13D05 Homological dimension and commutative rings Keywords:finite presentation; projective dimension; integral domain; valuation ring PDFBibTeX XMLCite \textit{D. L. Costa} and \textit{S.-E. Kabbaj}, Commun. Algebra 24, No. 3, 891--906 (1996; Zbl 0854.13013) Full Text: DOI References: [1] Bourbaki N., Commutative Algebra (1972) [2] Brewer J., Mich. Math. J 23 pp 33– (1976) · Zbl 0318.13007 · doi:10.1307/mmj/1029001619 [3] Costa D.L., Parametrizing families of non-Noetherian rings 22 pp 3997– (1994) · Zbl 0814.13010 [4] Dobbs D., Proc. A.M.S 56 pp 51– (1976) [5] Glaz S., Lecture Notes in Mathematics (1989) [6] Kaplansky I., Commutative Rings (1970) [7] Rotman J., An Introduction to Homological Algebra (1979) · Zbl 0441.18018 [8] Vasconcelos W., The Rings of Dimension Two (1976) · Zbl 0352.13003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.