TITLE: Standard example where Jacobson radical does not equal the nilradical
QUESTION [11 upvotes]: Is there a standard example in commutative algebra of a ring where the jacobson radical does not equal the nilradical?

REPLY [2 votes]: The formal power series ring $A[[x]]$, where $A$ is any Noetherian ring, is such an example.  You can show that $f = \sum_{n \geq 0}a_nx^n$ is  in the Jacobson radical if and only if $a_0$ is in the Jacobson radical of $A$, thus $x$ is in the Jacobson radical, but is not nilpotent.