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In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I 1 ⊆ I 2 ⊆ I 3 ⊆ ⋯ {\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots } of left ideals has a largest element; that is, there exists an n such that: I n = I n + 1 = ⋯ . {\displaystyle I_{n}=I_{n+1}=\cdots.}
Equivalently, a ring is left-Noetherian if every left ideal is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.
Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian , and many general theorems on rings rely heavily on Noetherian property.
Noetherian rings are named after Emmy Noether, but the importance of the concept was recognized earlier by David Hilbert, with the proof of Hilbert's basis theorem and Hilbert's syzygy theorem.