TITLE: Nil-Radical equals Jacobson Radical even though not every prime ideal is maximal?
QUESTION [15 upvotes]: Let's assume we have a commutative ring with identity. Can the Nil-Radical and the Jacobson Radical be equal in a non-trivial case (i.e. not every nonzero prime ideal in said ring is maximal)?
Are there any interesting examples of this case?

REPLY [4 votes]: Theorem 5.1 in T.Y. Lam's book "A First Course in Noncommutative Rings" states that every polynomial ring $R[T]$ over a commutative ring $R$ satisfies
$$rad \,\, R[T] = Nil(R[T]) = (Nil \,\,R)[T]$$