Dimension Theory of Noetherian Rings

2000 ◽  
pp. 129-147 ◽  
Author(s):  
T. H. Lenagan
Keyword(s):  
Noetherian Rings ◽  
Dimension Theory

Simple Noetherian Rings

1975 ◽  
Author(s):  
John Cozzens ◽  
CArl Faith
Keyword(s):  
Noetherian Rings

Localization in Noetherian Rings

1986 ◽  
Author(s):  
A. V. Jategaonkar
Keyword(s):  
Noetherian Rings

scholarly journals DIMENSION THEORY OF LINEAR SOLENOIDS

Fractals ◽  
2007 ◽  
Vol 15 (01) ◽  
pp. 63-72 ◽  
Author(s):  
JÖRG NEUNHÄUSERER
Keyword(s):  
Dimension Theory ◽  
Ergodic Measures ◽  
Full Dimension ◽  
Fractal Attractor

We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question of whether there exists a measure of full dimension.

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

Stone-čech Compactification and Dimension Theory for Regular σ-Frames

1989 ◽  
Vol s2-39 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Bernhard Banaschewski ◽  
Christopher Gilmour
Keyword(s):  
Dimension Theory

The converse to a well known theorem on Noetherian rings

1968 ◽  
Vol 177 (4) ◽  
pp. 278-282 ◽  
Author(s):  
Paul M. Eakin
Keyword(s):  
Noetherian Rings

scholarly journals Indecomposable modules over one-dimensional Noetherian rings

2007 ◽  
Vol 208 (2) ◽  
pp. 739-760 ◽  
Author(s):  
Meral Arnavut ◽  
Melissa Luckas ◽  
Sylvia Wiegand
Keyword(s):  
Noetherian Rings ◽  
Indecomposable Modules ◽  
One Dimensional

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

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