Question

Let R be a k-algebra where k is a field. Let K/k be a separable algebraic field extension.
Show that R is semiprime iff RK = R ⊗k K is semiprime.

Answer 1

The “if” part is clear (for any fieldextension), since, for any nilpotent ideal I ⊆ R, IK is a nilpotent ideal in RK.For the converse, assume that R is semiprime. Using standart argumentwith the lower nilradical replacing the Jacobson radical, we can deduce that RKis also semiprime.