An Artinian ring satisfies the descending chain condition on ideals. A commutative ring is Artinian exactly when it is Noetherian and has Krull dimension zero.
A commutative Noetherian ring is Artinian if and only if every prime ideal is maximal. Equivalently, it is Artinian exactly when its Krull dimension is zero.
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An Artinian ring is a type of ring in ring theory, which is a branch of abstract algebra. A ring \( R \) is called Artinian if it satisfies the descending chain condition (DCC) on ideals. This means that any descending chain of ideals in \( R \): \[ I_1 \supseteq I_2 \supseteq I_3 \supseteq \ldots \] eventually stabilizes, i.e.