Noetherian ring

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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 to be left-Noetherian or right-Noetherian respectively. Formally, every increasing sequence I 1 ⊆ I 2 ⊆ I 3 ⊆ ⋯ {\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots } of left (or right) ideals has a largest element; that is, there exists an n {\displaystyle n} such that I n = I n + 1 =...

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