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Answer to Question #24896 in Abstract Algebra for Hym@n B@ss

Question #24896
If R ⊆ Z(S), show that R ∩ Nil*(S) = Nil*(R).
Expert's answer
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).

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