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Math Help - localization of modules

  1. #1
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    localization of modules

    Hi there,

    I would like some help with the following:

    Let f be a homomorphism between two commutative rings A and B. S is a multiplicatively closed subset of A. Then f(S) is a multiplicatively closed subset of B. Now the book says that S(^-1)B and (f(S)^-1)B are isomorphic as S(^-1)A-modules.

    I get that we can see B as an A-module and from there that S(^-1)B is an S(^-1)A-module. But why is (f(S)^-1)B an S(^-1)A-module?

    J.
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  2. #2
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    Re: localization of modules

    Quote Originally Posted by Joolz View Post
    Hi there,

    I would like some help with the following:

    Let f be a homomorphism between two commutative rings A and B. S is a multiplicatively closed subset of A. Then f(S) is a multiplicatively closed subset of B. Now the book says that S(^-1)B and (f(S)^-1)B are isomorphic as S(^-1)A-modules.

    I get that we can see B as an A-module and from there that S(^-1)B is an S(^-1)A-module. But why is (f(S)^-1)B an S(^-1)A-module?

    J.
    why can we see B as an A-module? because we can define a \cdot b = f(a)b, for all a \in A and b \in B. so if s,t \in S and a \in A, b \in B, it is natural to define (s^{-1}a) \cdot (f(t))^{-1}b =(f(st))^{-1}f(a)b, which makes (f(S))^{-1}B an S^{-1}A-module.
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