##
**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.

Reviewer: T.Albu (Bucureşti)

### 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) |

### Keywords:

finite presentation; projective dimension; \((n,d)\)-ring; \(r\)-coherent ring; \(D+M\) ring
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\textit{D. L. Costa}, Commun. Algebra 22, No. 10, 3997--4011 (1994; Zbl 0814.13010)

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### References:

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[2] | DOI: 10.1307/mmj/1029001619 · Zbl 0318.13007 |

[3] | Dobbs D., Proc. A.M.S. 56 pp 51– (1976) |

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[6] | DOI: 10.1080/00927877508822059 · Zbl 0315.13010 |

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