Question

Show that:any commutative artinian ring is Hilbert.

Answer 1

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.]