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

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 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.

Equivariant K -Theory for Noetherian Rings

1989 ◽  
Vol s2-39 (3) ◽  
pp. 414-426 ◽  
Author(s):  
Timothy J. Hodges
Keyword(s):  
Noetherian Rings ◽  
K Theory

Some results concerning the local analytic branches of an algebraic variety

1953 ◽  
Vol 49 (3) ◽  
pp. 386-396 ◽  
Author(s):  
D. G. Northcott
Keyword(s):  
Algebraic Geometry ◽  
Power Series ◽  
Algebraic Variety ◽  
Recent Progress ◽  
Rapid Development ◽  
Noetherian Rings ◽  
Local Rings ◽  
Power Series Rings ◽  
Analytic Algebra ◽  
Topological Theories

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.

On Ramification Theory in Noetherian Rings

1959 ◽  
Vol 81 (3) ◽  
pp. 749 ◽  
Author(s):  
M. Auslander ◽  
D. A. Buchsbaum
Keyword(s):  
Noetherian Rings ◽  
Ramification Theory

scholarly journals Jacobson's conjecture and modules over fully bounded Noetherian rings

1974 ◽  
Vol 30 (1-3) ◽  
pp. 103-121 ◽  
Author(s):  
Arun Vinayak Jategaonkar
Keyword(s):  
Noetherian Rings

scholarly journals Constructing Gröbner bases for Noetherian rings

2013 ◽  
Vol 24 (2) ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Gröbner Basis ◽  
Finite Depth ◽  
Commutative Rings ◽  
Polynomial Ideal ◽  
Constructive Proof ◽  
Noetherian Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Prime Decomposition ◽  
Separate Paper

We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.

Dimension Theory of Noetherian Rings

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