Thread: 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.

Originally Posted by Joolz

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 , for all and . so if and , it is natural to define which makes an -module.