[SOLVED] R-modules and Homomorphism of Rings

[Sudharaka]

Hi everyone,

I find it difficult to get the exact meaning of the following question. What does "in a natural way" means? Is it that we have to show that there exist a isomorphism between \(S\) and a sub-module of \(R\)? Any ideas are greatly appreciated.

Question:

Given a ring homomorphism \(f:R\rightarrow S\), show that every \(S\)-module can be considered as an \(R\)-module in a natural way.

 

[Sudharaka]

Hi everyone,

I find it difficult to get the exact meaning of the following question. What does "in a natural way" means? Is it that we have to show that there exist a isomorphism between \(S\) and a sub-module of \(R\)? Any ideas are greatly appreciated.

Question:

Given a ring homomorphism \(f:R\rightarrow S\), show that every \(S\)-module can be considered as an \(R\)-module in a natural way.

I think I am getting some understanding. Suppose if we have a \(S\)-module called \(M\). Then we can define the operation \(R\times M\rightarrow M\) as, \(r.m=f( r)\,m\). Under this operation it's clear that \(M\) becomes a \(R\)-module. This can be verified by showing that \(M\) satisfies the \(R\)-module properties under the operation defined above.

This I believe is the natural way of defining an \(R\)-module from a given \(S\)-module where \(R\) and \(S\) are homomorphic. Correct me if I am wrong.

 

[Deveno]

Yes.

It is "natural" because it's pretty much the only way you have available to define

$r.m$.

This theorem tells us, for example, that any $\Bbb Z_n$-module can simply be regarded as a $\Bbb Z$-module, that is: an abelian group, and explains why reducing a polynomial (mod p) can be so useful in investigating polynomials with integer coefficients, because often information in the $S$-module can be "lifted" to an $R$-module.

 

[Sudharaka]

Yes.

It is "natural" because it's pretty much the only way you have available to define

$r.m$.

This theorem tells us, for example, that any $\Bbb Z_n$-module can simply be regarded as a $\Bbb Z$-module, that is: an abelian group, and explains why reducing a polynomial (mod p) can be so useful in investigating polynomials with integer coefficients, because often information in the $S$-module can be "lifted" to an $R$-module.

Thank you so much. I am sure your immense knowledge about these things will be of great help to me this semester.