Question

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

Answer 1

Under the assumption R ⊆ Z(S), any prime ideal p of S contractsto a prime p0 : = p ∩ R of R. Indeed, if a, b ∈ R are such that ab ∈p0, then aSb = abS ⊆p0S ⊆p, so we have, say, a ∈p ∩ R =p0. Therefore, any r ∈Nil*R lies in all primes p ofS, and so r ∈Nil*S. This gives R ∩ Nil*(S)= Nil*(R).