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.

please see attached pdf.

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

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