# Are all local rings Noetherian?

## Are all local rings Noetherian?

In commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.

## Is Noetherian a local property?

A commutative ring R can be non-Noetherian and have all of its localisations at prime ideals Noetherian, such as the infamous ∏∞i=1Z/2. So being Noetherian is not a local property.

How do you prove a ring is local?

A ring is called local if it has a unique maximal ideal. (a) Prove that a ring R with 1 is local if and only if the set of non-unit elements of R is an ideal of R. (b) Let R be a ring with 1 and suppose that M is a maximal ideal of R. Prove that if every element of 1+M is a unit, then R is a local ring.

Why are local rings called local?

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called “local behaviour”, in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

### Is being finitely generated a local property?

The property of being finitely generated is local.

### What is Noetherian induction?

From Encyclopedia of Mathematics. A reasoning principle applicable to a partially ordered set in which every non-empty subset contains a minimal element; for example, the set of closed subsets in some Noetherian space.

Is Za local ring?

A local ring is a ring with exactly one maximal ideal. The maximal ideal is often denoted \mathfrak m_ R in this case. We often say “let (R, \mathfrak m, \kappa ) be a local ring” to indicate that R is local, \mathfrak m is its unique maximal ideal and \kappa = R/\mathfrak m is its residue field.

How do you prove a local ring?

Proposition: Let A be a ring and m≠(1) an ideal of A such that every x∈A−m is a unit in A. Then A is a local ring and m its maximal ideal. Proof on book: every ideal ≠(1) consist of non-units, hence is contained in m. Hence m is the only maximal ideal of A.

#### Is a Noetherian ring finitely generated?

Since R is a Noetherian ring, J is a finitely generated ideal in R, and that finite generating set for J reduces to a generating set for J/I as an ideal of R/I.

#### Is a finitely generated?

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.

How do you prove a ring is Noetherian?

Proposition. If A is a Noetherian ring and f : A → B makes B an A-algebra so that B is a finitely generated A-module under the multiplication a.b = f(a)b, then B is a Noetherian ring.

What does Noetherian mean?

In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite length.

## Is polynomial ring local?

Any field or valuation ring is local. The ring of formal power series k[[X1…Xn]] over a field k or over any local ring is local. On the other hand, the polynomial ring k[X1…Xn] with n≥1 is not local.

## Why is it called a local ring?

Is Z X a Noetherian ring?

The ring Z[X,1/X] is Noetherian since it is isomorphic to Z[X, Y ]/(XY − 1).

Is every Artinian module Noetherian?

A module is Artinian (respectively Noetherian) if and only if it is so over its ring of homotheties. An infinite direct sum of non-zero modules is neither Artinian nor Noetherian. A vector space is Artinian (respectively Noetherian) if and only if its dimension is finite.

### What does finitely mean?

having bounds or limits; not infinite; measurable.

### Why are Noetherian rings important?

One big reason why they are important is that if R is noetherian the R[X] is also noetherian which then helps us see that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n …

Which of the following ring is neither Noetherian nor Artinian?

The ring F[X1,X2, . . . ] of polynomials over F in infinitely many variables is neither Artinian nor Noetherian. A descending chain of ideals that does not stabilize is constructed as in Example 3, and an ascending chain of ideals that does not stabilize is (X1) ⊂ (X1,X2) ⊂ (X1,X2,X3) ⊂ . . . . 5.

Is Za Noetherian ring?

(b) The ring Z is Noetherian: if we had a strictly increasing chain of ideals I0 ⊊ I1 ⊊ I2 ⊊ ททท in Z, then certainly I1 = 0, and thus I1 = (a) for some non-zero a ∈ Z. But there are only finitely many ideals in Z that contain I1 since they correspond to ideals of the finite ring Z/(a) by Lemma 1.21.