The Transfer of the Krull Dimension and the Gabriel Dimension to Subidealizers

1977 ◽  
Vol 29 (4) ◽  
pp. 874-888 ◽  
Author(s):  
Günter Krause ◽  
Mark L. Teply
Keyword(s):  
Krull Dimension ◽  
Dual Basis ◽  
Finitely Generated ◽  
Frequent Use ◽  
Gabriel Dimension

Let M be a right ideal of the ring T with identity. A unital subring R of T which contains M as a two-sided ideal is called a subidealizer ; the largest such subring is the idealizer I (M) of M in T. M is said to be generative if TM = T. In this case M is idempotent, and it follows from the dual basis lemma that T is finitely generated projective as a right R-module (see [7, Lemma 2.1]); we make frequent use of these two facts in this paper.

scholarly journals BOUNDED AND FULLY BOUNDED MODULES

2011 ◽  
Vol 84 (3) ◽  
pp. 433-440
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Krull Dimension ◽  
Finitely Generated ◽  
Morita Invariant ◽  
Prime Submodule ◽  
Invariant Properties ◽  
Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

scholarly journals Global dimensions of right coherent rings with left Krull dimension

1989 ◽  
Vol 39 (2) ◽  
pp. 215-223 ◽  
Author(s):  
Mark L. Teply
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Homological Dimension ◽  
Isomorphism Classes ◽  
Coherent Ring ◽  
Gabriel Dimension ◽  
Coherent Rings

The weak global dimension of a right coherent ring with left Krull dimension α ≥ 1 is found to be the supremum of the weak dimensions of the β-critical cyclic modules, where β < α. If, in addition, the mapping I → assl gives a bijection between isomorphism classes on injective left R-modules and prime ideals of R, then the weak global dimension of R is the supremum of the weak dimensions of the simple left R-modules. These results are used to compute the left homological dimension of a right coherent, left noetherian ring. Some analogues of our results are also given for rings with Gabriel dimension.

scholarly journals THE HOCHSCHILD COHOMOLOGY RING OF A CLASS OF SPECIAL BISERIAL ALGEBRAS

2010 ◽  
Vol 09 (01) ◽  
pp. 73-122 ◽  
Author(s):  
NICOLE SNASHALL ◽  
RACHEL TAILLEFER
Keyword(s):  
London Math ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Krull Dimension ◽  
Finitely Generated ◽  
Cohomology Rings ◽  
Hochschild Cohomology Ring ◽  
Special Biserial Algebras ◽  
Support Varieties

We consider a class of self-injective special biserial algebras ΛN over a field K and show that the Hochschild cohomology ring of dΛN is a finitely generated K-algebra. Moreover, the Hochschild cohomology ring of ΛN modulo nilpotence is a finitely generated commutative K-algebra of Krull dimension two. As a consequence the conjecture of [N. Snashall and Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc.88 (2004) 705–732], concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras.

scholarly journals A Remark on Coherent Overrings

1978 ◽  
Vol 21 (3) ◽  
pp. 373-375 ◽  
Author(s):  
Ira J. Papick
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
General Theorem ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Quotient Field ◽  
Valuation Rings ◽  
Finitely Presented

Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples.

On Weakly Laskerian and Weakly Cofinite Modules

2012 ◽  
Vol 19 (04) ◽  
pp. 693-698
Author(s):  
Kazem Khashyarmanesh ◽  
M. Tamer Koşan ◽  
Serap Şahinkaya
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cofinite Modules ◽  
Gorenstein Local Ring

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

scholarly journals Nonnoetherian geometry

2016 ◽  
Vol 15 (09) ◽  
pp. 1650176 ◽  
Author(s):  
Charlie Beil
Keyword(s):  
Commutative Algebra ◽  
Projective Dimension ◽  
Krull Dimension ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
One Dimensional ◽  
Simple Modules ◽  
Commutative Algebras

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of [Formula: see text] are all isomorphic if and only if [Formula: see text] is noetherian, if and only if the center [Formula: see text] of [Formula: see text] is noetherian, if and only if [Formula: see text] is a finitely generated [Formula: see text]-module. Furthermore, we show that [Formula: see text] is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

scholarly journals Some examples of modules over Noetherian rings

1982 ◽  
Vol 23 (1) ◽  
pp. 9-13 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Noetherian Ring ◽  
Essential Extension ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Finitely Generated

The purpose of this note is to prove the following result.Theorem 1. Let n be an integer greater than zero. There exists a prime Noetherian ring R of Krull dimension n + 1 and a finitely generated essential extension W of a simple R-module V suchthat(i) W has Krull dimension n, and(ii) W/V is n-critical and cannot be embedded in any of its proper submodules.We refer the reader to [6] for the definition and properties of Krull dimension.

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