Noetherian ring - Wikipedia
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring …
Noetherian Ring -- from Wolfram MathWorld
6 days ago · Noetherian Ring. A ring is called left (respectively, right) Noetherian if it does not contain an infinite ascending chain of left (respectively, right) ideals. In this case, the ring in …
A commutative ring R is called Noetherian if each ideal in R is nitely generated. This name honors Emmy Noether, who in her landmark paper [6] in 1921 proved proper-
We say that M is Noetherian if every submodule is nitely gener-ated. We say that a ring is Noetherian if it is Noetherian as a module over itself. Clearly every PID is Noetherian, since in …
Noetherian ring - Encyclopedia of Mathematics
Apr 3, 2024 · In particular, polynomial rings of the form $K [X_ {1} \dots X _ {n}] $ or $ \mathbf Z [ X _ {1} \dots X _ {n} ] $, where $K$ is a field and $\mathbf Z$ the ring of integers, and also …
23 Noetherian Rings. Definition 23.1. Noetherian R. ring is if every ideal in is finitely generated. In other words, every quotient can be obtained by imposing a finite number of relations. 23.1 …
Section 10.31 (00FM): Noetherian rings—The Stacks project
Noetherian rings. A ring R is Noetherian if any ideal of R is finitely generated. This is clearly equivalent to the ascending chain condition for ideals of R. By Lemma 10.28.10 it suffices to …
Noetherian module - Wikipedia
Noetherian module. In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by …
Theorem. Let Rbe a noetherian ring. If Mis a fg R-module then Mis R-noetherian. Proof. Suppose M= Xn i=1 Rxi. Let ˚: Rn!Mbe the R-epimorphism given by ei7!xi, where fe1;;engis the …
noetherian ring - PlanetMath.org
the ring of complex polynomials in two variables, where some ideals (the ideal generated by X X and Y Y, for example) are not principal, but all ideals are finitely generated. The Hilbert Basis …
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