Question

Let R be a k-algebra where k is a field. Let K/k be a separable algebraic field extension.
Show that Nil*(RK) = (Nil*(R))K.

Answer 1

Since Nil*(R) ⊆Nil*(RK), we have (Nil*(R))K ⊆Nil*(RK). On the other hand,RK/(Nil*R)K ∼ (R/Nil*R)K
is semiprime, so (Nil*(R))K⊇ Nil*(RK). Therefore, the desired equality follows.