• Corpus ID: 14597668

Trivial Extensions of Local Rings and a Conjecture of Costa

@article{Kabbaj2002TrivialEO,
  title={Trivial Extensions of Local Rings and a Conjecture of Costa},
  author={S. Kabbaj and Najib Mahdou},
  journal={arXiv: Commutative Algebra},
  year={2002}
}
This paper partly settles a conjecture of Costa on (n,d)-rings, i.e., rings in which n-presented modules have projective dimension at most d. For this purpose, a theorem studies the transfer of the (n,d)-property to trivial extensions of local rings by their residue fields. It concludes with a brief discussion -backed by original examples- of the scopes and limits of our results. 
View PDF on arXiv
30 Citations
Background Citations
8
Methods Citations
2

30 Citations

Citation Type

Citation Type

Gorenstein dimensions in trivial ring extensions
In this paper, we show that the Gorenstein global dimension of trivial ring extensions is often infinite. Also we study the transfer of Gorenstein properties between a ring and its trivial ring…
MATLIS’ SEMI-REGULARITY IN TRIVIAL RING EXTENSIONS OF INTEGRAL DOMAINS
This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis’) semi-regular…
Some Homological Properties of Subring Retract and Applications to Fixed Rings
Abstract In this paper we study the relationship between some homological properties such as the weak and the global dimension, the strong n-coherence, and the (n, d)-property, where n and d are two…
Matlis' semi-regularity in trivial ring extensions issued from integral domains
This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis') semi-regular…
Matlis' semi-regularity in trivial ring extensions
This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis') semi-regular…
Commentationes Mathematicae Universitatis Carolinae
In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if R ⊆ T is a ring extension such that mT ⊆ R for some regular…
On n-flat modules and n-Von Neumann regular rings
  • N. Mahdou
  • Mathematics
    Int. J. Math. Math. Sci.
  • 2006
TLDR
It is shown that in a class of principal rings and aclass of local Gaussian rings, a weak n-Von Neumann regular ring is a (CH)-ring.
On Gorenstein global dimension in trivial ring extensions
In this paper, we compare the Gorenstein homological dimension of a ring $R$ and of its trivial ring extension by an module $E$.
On Weakly Semihereditary Rings
In this paper, we introduce a generalization of the well-known notion of a semihereditary ring which we call a weak semihereditary ring. We investigate the transfer of the weak semihereditary…
Combinatorial Dimensions: Indecomposability on Certain Local Finite Dimensional Trivial Extension Algebras
We study problems related to indecomposability of modules over certain local finite dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of…
...
...

References

SHOWING 1-10 OF 23 REFERENCES
On the Global Dimensions of D+M
  • D. E. Dobbs
  • Mathematics
    Canadian Mathematical Bulletin
  • 1975
This note answers affirmatively a question of the author [4, p. 456], by producing an example of an integrally closed quasi-local non valuation domain of global dimension 3, each of whose overrings…
Commutative extensions by canonical modules are Gorenstein rings
Reiten has demonstrated that the trivial Hochschild extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein local ring. Here it is proved that any commutative extension of a…
When is $D+M$ coherent?
Let V be a valuation ring of the form K + M, where K is a field and M(# 0) is the maximal ideal of V. Let D be a proper subring of K. Necessary and sufficient conditions are given that the ring D + M…
On the homotopy Lie algebra of a local ring
Parameterizing families of non-noetherian rings
(1994). Parameterizing families of non-noetherian rings. Communications in Algebra: Vol. 22, No. 10, pp. 3997-4011.
ON COSTA'S CONJECTURE
Fn ! Fn 1 ! ! F0 ! M ! 0 of R-modules in which each Fi is finitely generated and free. In particular, 0-presented and 1-presented R-modules are finitely generated and finitely presented R-modules…
Commutative Coherent Rings
Preliminaries.- to coherence.- Fundamental concepts.- Ring extensions.- Ring constructions and overrings.- Particular coherent rings.- Polynomial rings.- Coherent algebras.
Explicit formulae for the global homological dimensions of trivial extensions of rings
General Néron desingularization
Commutative torsion stable rings
...
...