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Parameterizing families of non-Noetherian rings. (English) Zbl 0814.13010

All rings considered in this paper are supposed to be commutative with identity. Let \(R\) be a ring and \(M\) an \(R\)-module. \(M\) is said to be \(n\)- presented, where \(n\) is a positive integer, if there exists an exact sequence \(F_ n \to F_{n-1} \to \cdots \to F_ 0 \to M \to 0\) in which each \(F_ i\) is a finitely generated free \(R\)-module. For any integer \(n \geq 0\) one defines the \(n\)-dimension \(n\)-\(\dim (R)\) of \(R\) to be the supremum of the projective dimensions pd\(_ R (M)\) for all \(n\)- presented \(R\)-modules \(M\). Finally, if \(n,d \geq 0\) are integers, then \(R\) is said to be an \((n,d)\)-ring if \(n\)-\(\dim(R) \leq d\). The main purpose of this paper is to investigate these rings. Some important classes of commutative rings are \((n,d)\)-rings: semisimple rings are exactly the (0,0)-rings, von Neumann regular rings are exactly the (1,0)-rings, hereditary rings are exactly the (0,1)-rings, semihereditary rings are exactly the (1,1)-rings, etc. Among others, the property of a ring \(R\) to be an \((n,d)\)-ring is studied in relationship to that of being \(r\)- coherent, local-global questions on \((n,d)\)-rings are investigated. \(D + M\) constructions related to \((n,d)\)-rings are performed, and some descent results concerning \((n,d)\)-rings are established.

MSC:

13D05 Homological dimension and commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
18G20 Homological dimension (category-theoretic aspects)
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References:

[1] Bourbaki N., Commutative Algebra (1972)
[2] DOI: 10.1307/mmj/1029001619 · Zbl 0318.13007
[3] Dobbs D., Proc. A.M.S. 56 pp 51– (1976)
[4] Kaplansky I., Commutative Rings (1970)
[5] Rotman J., An Introduction to Homological Algebra (1979) · Zbl 0441.18018
[6] DOI: 10.1080/00927877508822059 · Zbl 0315.13010
[7] Vasconcelos W., The Rings of Dimension Two (1976) · Zbl 0352.13003
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