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Answer to Question #22748 in Algebra for Tsit Lam

Question #22748
Show that:any commutative artinian ring is Hilbert.
Expert's answer
Let R be a commutativeartinian ring, and consider any quotient R' of R. Then R' isalso artinian, and so rad(R') = Nil(R'). Then R isHilbert. [Alternatively, we can use thewell-known fact that R, being artinian, has Krull-dimension 0.) Thismeans that any prime ideal of R is maximal, so R is clearlyHilbert.]

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