Let $L/K$ be an extension of fields, consider the commutative rings $A = K[X_1, \ldots, X_m]$ and $B = L[X_1, \ldots, X_m]$ for some natural number $m$.
If $I$ is a prime ideal of $A$, then certainly $IB$ (smallest ideal of $B$ containing $I$) need not be prime, even if $m = 1$: an irreducible univariate polynomial can become reducible over a finite field extension.
But can the converse happen? Could $I$ be an ideal of $A$ which is not prime such that $IB$ is prime? Similarly, can $IB$ be maximal if $I$ is not maximal?