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# Noetherian module

A (left or right) module $M$ over a ring $R$ is said to be *Noetherian*
if the following equivalent conditions hold:

1. Every submodule of $M$ is finitely generated over $R$.

2. The ascending chain condition holds on submodules.

3. Every nonempty family of submodules has a maximal element.

For example, the $\mathbb{Z}$-module $\mathbb{Q}$ is not Noetherian, as it is not finitely generated, but the $\mathbb{Z}$-module $\mathbb{Z}$ is Noetherian, as every submodule is generated by a single element.

Observe that changing the ring can change whether a module is Noetherian or not: for example, the $\mathbb{Q}$-module $\mathbb{Q}$ is Noetherian, since it is simple (has no nontrivial submodules).

There is also a notion of Noetherian for rings: a ring is left Noetherian if it is Noetherian as a left module over itself, and right Noetherian if it is Noetherian as a right module over itself. For non-commutative rings, these two notions can differ.

The corresponding property for groups is usually called the maximal condition.

Finally, there is the somewhat related notion of a Noetherian topological space.

## Mathematics Subject Classification

13E05*no label found*33C75

*no label found*33E05

*no label found*14J27

*no label found*86A30

*no label found*14H52

*no label found*

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## Comments

## new line

backslash n: \n

foreslash n: /n

that is all.

## Equivalent Conditions

I'm just wondering if the first sentence shouldn't be "if <<any>> of the following equivalent conditions hold." Thanks a lot.

Regards,

Keenan

## Re: Equivalent Conditions

It is preferable if these sorts of issues were filed as corrections (in this case, meta/minor). Sifting through posts attached to an entry can be a difficult matter, but (hopefully) sifting through unaddressed corrections attached to an entry is not.

## Re: Equivalent Conditions

No problem; sorry about that.