# Modules

• December 27th 2011, 02:44 PM
jcir2826
Modules
Let R and S be commutative rings and let phi: R to S be a ring homomorphism.

If M is an S-Module, prove that M is also and R-Module if we define rm = phi(r)m.
for all r in R and all m in M.

• December 27th 2011, 10:56 PM
FernandoRevilla
Re: Modules
Quote:

Originally Posted by jcir2826
Let R and S be commutative rings and let phi: R to S be a ring homomorphism.If M is an S-Module, prove that M is also and R-Module if we define rm = phi(r)m. for all r in R and all m in M.

It is an easy consequence of the corresponding definitions:

$(i)\;\;(r_1+r_2)m=\phi(r_1+r_2)m=(\phi(r_1)+\phi(r _2))m=\phi(r_1)m+\phi(r_2)m=$

$r_1m+r_2m\quad (\forall r_1,r_2\in R,\;\forall m\in M)$

...

$(iv)\;\; 1m=\phi(1)m=1m=m\quad (\forall m\in M)$

Try the rest.