scholarly journals Presentations of rings with a chain of semidualizing modules

2017 ◽  
Vol 121 (2) ◽  
pp. 161
Author(s):  
Ensiyeh Amanzadeh ◽  
Mohammad T. Dibaei
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Semidualizing Modules ◽  
Dualizing Module ◽  
A Chain

Inspired by Jorgensen et al., it is proved that if a Cohen-Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$-modules of length $n$, then $R\cong Q/(I_1+\cdots +I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\dots , I_n$ of $Q$; and, for each $\Lambda \subseteq [n]$, the ring $Q/(\sum _{\ell \in \Lambda } I_\ell )$ has some interesting cohomological properties. This extends the result of Jorgensen et al., and also of Foxby and Reiten.

On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions

2015 ◽  
Vol 58 (1) ◽  
pp. 134-143
Author(s):  
Saeed Nasseh
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Ring Extensions

AbstractWe show that under some conditions a Gorenstein ring R satisfies the Generalized Auslander–Reiten conjecture if and only if R[x] does. When R is a local ring we prove the same result for some localizations of R[x].

A Gorenstein Ring with Larger Dilworth Number than Sperner Number

2000 ◽  
Vol 43 (1) ◽  
pp. 100-104 ◽  
Author(s):  
James S. Okon ◽  
J. Paul Vicknair
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Embedding Dimension ◽  
Gorenstein Rings

AbstractA counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension 3 with larger Dilworth number than Sperner number. The Dilworth number of is computed when A is an unramified principal Artin local ring.

scholarly journals Almost Gorenstein rings arising from fiber products

2020 ◽  
pp. 1-18
Author(s):  
Naoki Endo ◽  
Shiro Goto ◽  
Ryotaro Isobe
Keyword(s):  
Local Ring ◽  
The Other ◽  
Gorenstein Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Gorenstein Rings ◽  
Fiber Product

Abstract The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

scholarly journals A Note on Gorenstein Rings of Embedding Codimension Three

1973 ◽  
Vol 50 ◽  
pp. 227-232 ◽  
Author(s):  
Junzo Watanabe
Keyword(s):  
Local Ring ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Complete Intersections ◽  
Gorenstein Ring ◽  
Regular Local Ring ◽  
Gorenstein Rings ◽  
Number Of Generators ◽  
Five Elements

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

Relative and Tate homology with respect to semidualizing modules

2014 ◽  
Vol 13 (08) ◽  
pp. 1450058 ◽  
Author(s):  
Zhenxing Di ◽  
Xiaoxiang Zhang ◽  
Zhongkui Liu ◽  
Jianlong Chen
Keyword(s):  
Exact Sequence ◽  
London Math ◽  
Projective Dimension ◽  
Relative Cohomology ◽  
Relative Homology ◽  
Semidualizing Module ◽  
Coherent Ring ◽  
Semidualizing Modules ◽  
Tate Homology ◽  
Dualizing Module

We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate ℱC-resolutions, where C is a semidualizing module. We show firstly that the class of modules admitting a Tate ℱC-resolution is equal to the class of modules of finite 𝒢(ℱC)-projective dimension. Then an Avramov–Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather–Wagstaff et al. [Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010) 571–600], we establish a balance result for such Tate homology over a Cohen–Macaulay ring with a dualizing module by using a good conclusion provided in [E. E. Enochs, S. E. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. London Math. Soc. 44 (2012) 439–442].

A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory

2017 ◽  
Vol 120 (2) ◽  
pp. 161 ◽  
Author(s):  
Tony J. Puthenpurakal
Keyword(s):  
Local Ring ◽  
Representation Type ◽  
Gorenstein Ring ◽  
Stable Category ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Isomorphism Classes ◽  
Liaison Theory ◽  
Gorenstein Local Ring ◽  
Infinite Set

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\operatorname{\underline{CM}}(A)$ be the stable category of maximal Cohen-Macauley $A$-modules and let $\operatorname{\underline{ICM}}(A)$ denote the set of isomorphism classes in $\operatorname{\underline{CM}}(A)$. We define a function $\xi \colon \operatorname{\underline{ICM}}(A) \to \mathbb{Z}$ which behaves well with respect to exact triangles in $\operatorname{\underline{CM}}(A)$. We then apply this to (Gorenstein) liaison theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liaison classes of $\{\,\mathfrak{m}^n \mid n\geq 1 \,\}$ is an infinite set. We also prove that if $A$ is Henselian with finite representation type with $A/\mathfrak{m}$ uncountable then for each $m \geq 1$ the set $\mathcal {C}_m = \{\, I \mid I \text { is a codim $2$ CM-ideal with } e_0(A/I) \leq m \,\}$ is contained in finitely many even liaison classes $L_1,\dots ,L_r$ (here $r$ may depend on $m$).

scholarly journals Chains of semidualizing modules

2018 ◽  
Vol 17 (07) ◽  
pp. 1850118
Author(s):  
Ensiyeh Amanzadeh
Keyword(s):  
Local Ring ◽  
Noetherian Local Ring ◽  
Bass Numbers ◽  
Semidualizing Modules ◽  
Insight Into

Let [Formula: see text] be a commutative Noetherian local ring. We study the suitable chains of semidualizing [Formula: see text]-modules. We prove that when [Formula: see text] is Artinian, the existence of a suitable chain of semidualizing modules of length [Formula: see text] implies that the Poincar[Formula: see text] series of [Formula: see text] and the Bass series of [Formula: see text] have very specific forms. Also, in this case, we show that the Bass numbers of [Formula: see text] are strictly increasing. This gives an insight into the question of Huneke about the Bass numbers of [Formula: see text].

Complexes acyclic up to integral closure

1994 ◽  
Vol 116 (3) ◽  
pp. 401-414
Author(s):  
D. Katz
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Noetherian Ring ◽  
Integral Closure ◽  
Koszul Complex ◽  
Gorenstein Ring ◽  
Tight Closure ◽  
Commutative Noetherian Ring ◽  
Remarkable Result ◽  
Depth Sensitivity

In [R] D. Rees introduced the notions of reduction and integral closure for modules over a commutative Noetherian ring and proved the following remarkable result. Let R be a locally quasi-unmixed Noetherian ring and I an ideal generated by n elements. Suppose that height (I) = h. Then the ith module of cycles in the Koszul complex on a set of n generators for I is contained in the integral closure of the ith module of boundaries for i > n − h. This result should be considered a dimension-theoretic analogue of the famous depth sensitivity property of the Koszul complex demonstrated by Serre and Auslander-Buschsbaum in the 1950s. At roughly the same time, Hoschster and Huneke introduced the notion of tight closure and thereafter gave a number of theorems in the same (though considerably broader) vein for tight closure. In particular, in [HH] they showed that if R is an equidimensional local ring of characteristic p > 0, which is a homomorphic image of a Gorenstein ring, then for all i > 0, the ith module of cycles is contained in the tight closure of the ith module of boundaries for any complex satisfying the so-called standard rank and height conditions (see the definitions below). Since the tight closure is contained in the integral closure for such rings, the result of Hochster and Huneke extends (in characteristic p) considerably the result of Rees. In fact, their result could be considered a dimension-theoretic analogue of the Buchsbaum-Eisenbud exactness theorem ([BE]), which in a certain sense is the ultimate depth sensitivity theorem. Moreover, using the technique of reduction to characteristic p, Hochster and Huneke have shown that their results hold in equicharacteristic zero as well, whenever the tight closure is defined.

On coefficient ideals

2018 ◽  
Vol 167 (02) ◽  
pp. 285-294 ◽  
Author(s):  
R. CALLEJAS-BEDREGAL ◽  
V. H. JORGE PÉREZ ◽  
M. DUARTE FERRARI
Keyword(s):  
Local Ring ◽  
Structure Theorem ◽  
Noetherian Local Ring ◽  
Analytic Spread ◽  
A Chain ◽  
Primary Ideals

AbstractLet (R, 𝔪) be a Noetherian local ring and I an arbitrary ideal of R with analytic spread s. In [3] the authors proved the existence of a chain of ideals I ⊆ I[s] ⊆ ⋅⋅⋅ ⊆ I[1] such that deg(PI[k]/I) < s − k. In this article we obtain a structure theorem for this ideals which is similar to that of K. Shah in [10] for 𝔪-primary ideals.

scholarly journals RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250049 ◽  
Author(s):  
F. ALINIAEIFARD ◽  
M. BEHBOODI
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Gorenstein Ring ◽  
Isomorphism Classes ◽  
Artinian Rings ◽  
Vertex Set ◽  
Positive Genus

Let R be a commutative ring and 𝔸(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(𝔸𝔾(R)). It is shown that if R is an Artinian ring such that γ(𝔸𝔾(R)) < ∞, then either R has only finitely many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v.dimR/𝔪𝔪/𝔪2= 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(𝔸𝔾(R)) = g and (ii) |R/𝔪| ≤ q for every maximal ideal 𝔪 of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(𝔸𝔾(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

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