# Commutative ring proofs

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.

Please see attached pdf.

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.