Center of noetherian rings

Question

Is it true that the center of a right Noetherian ring (with identity) is always a Noetherian ring ?

Answer 1

No, this is not true, even under some rather restrictive conditions on the ring. A number of counterexamples, showing that if $R$ is a prime Noetherian PI ring, the center of $R$ need not be Noetherian can be found in Examples 5.1.16 through 5.1.18 of L. Rowen's book Polynomial identities in ring theory. Additional examples are sketched in Exercises 1 and 2 of §5.1 in the same book.