Answer to Question #22735 in Abstract Algebra for Irvin

Question

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

Answer 1

This inclusion in not true ingeneral.If we take R = Z and S=Z2Z.
Here, rad S^−1R =2Z₍₂₎and R ∩ rad S^−1R =2Z, but rad R =0.