ABSTRACT
One of the fundamental constructions of commutative ring theory is that of the classical ring of quotients at a multiplicatively-closed subset of the ring. Specifically, if R is a commutative ring and S is a nonempty multiplicatively-closed subset of R not containing 0 then one can construct the ring S−1R by the following sequence of steps:
Define an equivalence relation ~ on the set R × S by setting (r, s) ~ (r′, s′) if and only if there exists an element s″ in S such that s″(s′r - sr’) = 0. Denote the equivalence class of the pair (r, s) by r/s and denote the set of all such equivalence classes by S−1R.
Define addition and multiplication on S−1R as follows: if r/s and r′/s′ are elements of S−1R, then r/s + r′/s′ = [s′r + sr′]/ss′ and (r/s)·(r′/s′) = rr′/ss′.
