# 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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