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    Commutative ring proofs

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    See the attached file.
    Let R be any commutative ring and S a subset of R n f0g containing no zero divisors.
    Let X be the Cartesian product R  S and denote a relation  on X where (a; b)  (c; d).
    (a) Show that  is an equivalence relation on X.
    (b) Denote the equivalence class of (a; b) by a=b and the set of equivalence classes by RS (called the
    localization of R at S). Show that RS is a commutative ring with 1.
    (c) If a 2 S show that fra=a: r 2 Rg is a subring of RS and that r 7! ra=a is a monomorphism, so that
    R can be identified with a subring with RS.
    (d) Show that every s 2 S is a unit in RS.
    (e) Give a universal" definition for the ring RS and show that RS is unique up to isomorphism.

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    I need a detailed, rigorous proof of this with explanation of the steps so I can learn.

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    The extension F is unique; that is, if G : Q --> T ...

    Solution Summary

    This provides examples of proofs regarding commutative rings and equivalence relations and classes, subrings and monomorphisms, units, and isomorphism.