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Math Help - Ring

  1. #1
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    Ring

    Let R be a ring, and I be an ideal of G.
    Show that there is a one to one and onto correspondence between the subrings of R/I and those subrings of R which contain I.
    Hint: maybe the "natural" homomorphism f: R-> R/I defined by f(x)= x+I

    can someone help me please
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  2. #2
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    Quote Originally Posted by evonnova View Post
    Let R be a ring, and I be an ideal of G.
    Show that there is a one to one and onto correspondence between the subrings of R/I and those subrings of R which contain I.
    Hint: maybe the "natural" homomorphism f: R-> R/I defined by f(x)= x+I

    can someone help me please
    Saying "one-to-one and onto correspondence" is not necessary because a one-to-one correspondence is defined as a one-to-one and onto.

    Define \pi: R\mapsto R/I to be the natural projection. If S is a subring of R then \pi (S) is a subring of R/I. If K is a subring of R/I then \pi^{-1} (K) is a subring of R which contains R.

    Let X be the set of all subrings of R which contain I and Y be the set of all subrings of R/I. By above paragraph define \hat \pi: X\mapsto Y as \hat \pi (x) = \pi (x) for all x\in X.
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