Question

Let R be a commutative domain and S^(−1)R be the localization of R at a multiplicative set S.
Is "rad R ⊆ R ∩ rad S−1R"?

Answer 1

This inclusion in not true ingeneral. Let’s take a commutative local domain (R,m) with nonzero m. Then rad R
= m, but for S = R {0}, S^−1R is a field, with rad S^−1R = 0.