Abstract Algebra Answers

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.

Let R,K be algebras over a commutative ring k such that R is k projective and K ⊇ k.
Show that R ∩ Nil*(R ⊗k K) = Nil*(R).

If R ⊆ Z(S), show that R ∩ Nil*(S) = Nil*(R).

Let R ⊆ S be rings. Show that R ∩ Nil*(S) ⊆ Nil*(R).

Show that Nil*R is precisely the set of all strongly nilpotent elements of R.

For any ideal A in a ring R, show that √A consists of s ∈ R such that every n-system containing s meets A.

Show that if the ideals in R satisfy ACC (e.g. when R is left noetherian), then R has only finitely many minimal prime ideals.

Show that any prime ideal p in a ring R contains a minimal prime ideal. Using this, show that the lower nilradical Nil*R is the intersection of all the minimal prime ideals of R.

Let I be a left ideal in a ring R such that, for some integer n, an = 0 for all a ∈ I. Show that I ⊆ Nil*R.

Let I be a left ideal in a ring R such that, for some integer n, an = 0 for all a ∈ I.
Show that I contains a nonzero nilpotent left ideal, and R has a nonzero nilpotent ideal.