# Answer to Question #24899 in Abstract Algebra for jeremy

Question #24899

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.

Show that Nil*(RK) = (Nil*(R))K.

Expert's answer

Since Nil*(

is semiprime, so (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.
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