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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 question is said to satisfy the ascending chain condition on left (respectively, right) ideals.

A ring is said to be Noetherian if it is both left and right Noetherian. For a ring R, the following are equivalent:

1. R satisfies the ascending chain condition on ideals (i.e., is Noetherian).

2. Every ideal of R is finitely generated.

3. Every set of ideals contains a maximal element.


See also

Artinian Ring, Ascending Chain Condition, Left Ideal, Local Ring, Noether-Lasker Theorem, Noetherian Module, Right Ideal

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References

Hungerford, T. W. Algebra, 8th ed. New York: Springer-Verlag, 1997.

Referenced on Wolfram|Alpha

Noetherian Ring

Cite this as:

Weisstein, Eric W. "Noetherian Ring." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NoetherianRing.html

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