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.

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.

... This provides examples of proofs regarding subfields and prime fields, polynomial in a ... Let R be a commutative ring with 1. Prove that f (x) = a0 + a1 x ...Proof. ...

... This solution provides the detailed proofs and examples of ... Proposition 1. A commutative ring with 1 ≠ 0 is ...Proof: (⟹) Suppose that is a local ring with a ...

Ring and ideal proofs. ** Please see the attached file for the complete problem description **. ... (c) The nilradical ideal nil(R) of a commutative ring R with 1 is ...

... be the power set of . Define addition and multiplication in R as follows: Proof: Let. Let's first show that is a commutative ring. (i) Let , , . Then we have. ...

... provides examples of abstract algebra proofs regarding distributive law, commutative rings, integral domains ... the defining rules of a ring tells us ...