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Math Help - Modules

  1. #1
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    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.

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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Modules

    Quote Originally Posted by jcir2826 View Post
    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.
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