Abstract Algebra Answers

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.

Let I be a left ideal in a ring R such that, for some integer n ≥ 2, an = 0 for all a ∈ I. Show that an−1Ran−1 = 0 for all a ∈ I.

For any ring R and any ordinal α, define Nα(R) as follows. For α = 1, N1(R) is a nil subideal of Nil*R which contains all nilpotent one-sided ideals of R. If α is the successor of an ordinal β, define
Nα(R) = {r ∈ R : r + Nβ(R) ∈ N1 (R/Nβ(R))}. If α is a limit ordinal, define Nα(R) = (Union over β<α) Nβ(R).
Show that Nil*R = Nα(R) for any ordinal α with Card α > Card R.