Reference no: EM131027708

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 denote 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 identified with a subring with RS.

(d) Show that every s 2 S is a unit in RS.

(e) Give a universal" definition for the ring RS and show that RS is unique up to isomorphism.