Noetherian ring

Builds on Ascending chain condition Ideal generated by a subset Commutative ring

Definition

Let $R$ be a commutative ring. The ring $R$ is Noetherian if the set of its ideals, ordered by inclusion, satisfies the ascending chain condition. That is, for every chain $I_1 \subseteq I_2 \subseteq \dots$ of ideals of $R$ , there is $N$ with $I_n = I_N$ for all $n \ge N$ .

Remarks

The ring $R$ is Noetherian if and only if every ideal of $R$ is finitely generated. Every field and every principal ideal domain is Noetherian. By Hilbert's basis theorem, $R[x_1, \dots, x_n]$ is Noetherian when $R$ is. The polynomial ring in infinitely many variables $x_1, x_2, \dots$ over a field is not Noetherian: $(x_1) \subset (x_1, x_2) \subset \dots$ never stops. For a noncommutative ring, one defines left and right Noetherian rings with left and right ideals.

Reading path

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Ascending chain condition definition
Group concept
Ring definition
Commutative ring definition
Ideal definition
Ideal generated by a subset definition
Noetherian ring definition

Tag	`0MVH4Y1CRZ`
URI	`m2://0MVH4Y1CRZ`
Kind	definition
Writing	W1 stated
MSC	13E05
Defines	Noetherian ring
Uses	Ascending chain condition, Ideal generated by a subset, Commutative ring

Noetherian ring

Definition

Remarks

Reading path

Where this leads

Further reading

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