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    Example of an Artinian ring

    P-TO35W

    Area
    Algebra
    Topics
    Noetherian Rings, Rings
    Status
    draft

    Give an example of an Artinian ring.

    Solution

    For any integer \(n\ge2\), the ring \[ \ZZ/n\ZZ \] is Artinian.

    Indeed, it is finite. Any descending chain of ideals \[ I_1\supseteq I_2\supseteq I_3\supseteq\cdots \] is therefore a descending chain of subsets of a finite set, so there can be only finitely many strict inclusions. Hence the chain stabilizes.

    Thus every finite ring is Artinian, and \(\ZZ/n\ZZ\) is a basic example.

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