# Answer on Abstract Algebra Question for jeremy

Question #24898

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.

Show that R is semiprime iff RK = R ⊗k K is semiprime.

Expert's answer

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*RK*is also semiprime.Need a fast expert's response?

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