# Is every Artinian module Noetherian?

## 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.

## Are Noetherian rings Artinian?

A ring A is noetherian, respectively artinian, if it is noetherian, respectively artinian, considered as an A-module. In other words, the ring A is noetherian, respectively artinian, if every chain a1 ⊆ a2 ⊆ ··· of ideal ai in A is stable, respectively if every chain a1 ⊇ a2 ⊇··· of ideals ai in A is stable.

Are Artinian modules finitely generated?

A quotient of an Artinian ring (by a two sided ideal) is Artinian. A finitely generated module over an Artinian ring is Artinian.

### What is Noetherian R 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 inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules.

### Who is Noetherian?

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.

Why is Z Noetherian?

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. Hence the chain cannot be infinitely long, and thus Z is Noetherian.

#### Is every finite ring an Artinian ring?

Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. This implies that a simple ring is left Artinian if and only if it is right Artinian.

#### What does Artinian mean?

Filters. (algebra, of a ring) In which any descending chain of ideals eventually starts repeating. adjective. (algebra, of a module) In which any descending chain of submodules eventually starts repeating.

Can we prove every submodule of a Noetherian module is finitely generated How?

Proof:- Let M be noetherian R-module and N be a submodule of M. Since M is noetherian means every submodule N of M is finitely generated. Let L be a submodule of N then L is submodule of M. Thus, L is finitely generated and so N is noetherian module.

## Is a Noetherian module finitely generated?

Every submodule of a noetherian module is noetherian. Since M is noetherian means every submodule N of M is finitely generated. Let L be a submodule of N then L is submodule of M. Thus, L is finitely generated and so N is noetherian module.

## What is Noetherian domain?

Any principal ideal ring, such as the integers, is Noetherian since every ideal is generated by a single element. This includes principal ideal domains and Euclidean domains. A Dedekind domain (e.g., rings of integers) is a Noetherian domain in which every ideal is generated by at most two elements.

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 …

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