# Answer to Question #22735 in Abstract Algebra for Irvin

Question #22735

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" ?

Is "R ∩ rad S−1R ⊆ rad R" ?

Expert's answer

This inclusion in not true ingeneral.If we take R = Z and S=Z2Z.

Here, rad S

Here, rad S

^{−1}R =2Z_{(2)}and R ∩ rad S^{−1}R =2Z, but rad R =0.
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