The Transfer of the Krull Dimension and the Gabriel Dimension to Subidealizers* : Erratum

1978 ◽  
Vol 30 (03) ◽  
pp. 672
Author(s):  
Günter Krause ◽  
Mark L. Teply
Keyword(s):  
Krull Dimension ◽  
Gabriel Dimension

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.

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.

Krull Dimension and Gabriel Dimension of Rings and Modules

1987 ◽  
pp. 143-170
Author(s):  
Constantin Nǎstǎsescu ◽  
Freddy van Oystaeyen
Keyword(s):  
Krull Dimension ◽  
Gabriel Dimension

scholarly journals Krull dimension-nilpotency and Gabriel dimension

1973 ◽  
Vol 79 (4) ◽  
pp. 716-720 ◽  
Author(s):  
Robert Gordon ◽  
Thomas H. Lenagan ◽  
J. C. Robson
Keyword(s):  
Krull Dimension ◽  
Gabriel Dimension

On Maximal Torsion Radicals, II

1975 ◽  
Vol 27 (1) ◽  
pp. 115-120 ◽  
Author(s):  
John A. Beachy
Keyword(s):  
Associative Ring ◽  
Commutative Rings ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Previous Theorem ◽  
Maximal Torsion ◽  
Gabriel Dimension ◽  
Right Noetherian Rings ◽  
Minimal Prime

Let R be an associative ring with identity, and let denote the category of unital left R-modules. The Walkers [6] raised the question of characterizing the maximal torsion radicals of , and showed that if R is commutative and Noetherian, then there is a one-to-one correspondence between maximal torsion radicals and minimal prime ideals of R [6, Theorem 1.29]. Popescu announced [5, Theorem 2.5] that the result remains valid for commutative rings with Gabriel dimension (in the terminology of [2]). Theorem 4.6 below shows that the result holds for rings (not necessarily commutative) with Krull dimension on either the left or right, extending the previous theorem for right Noetherian rings which appeared in [1].

Krull Dimension and Gabriel Dimension of an Ordered Set

1987 ◽  
pp. 121-142
Author(s):  
Constantin Nǎstǎsescu ◽  
Freddy van Oystaeyen
Keyword(s):  
Ordered Set ◽  
Krull Dimension ◽  
Gabriel Dimension

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.

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