Torsion and Torsion free Modules

Le résumé fourni par la source

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 −1 R 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 −1 R. Define addition and multiplication on S −1 R as follows: if r/s and r′/s′ are elements of S −1 R, then r/s + r′/s′ = [s′r + sr′]/ss′ and (r/s)·(r′/s′) = rr′/ss′.

Ce résumé expose les affirmations des auteurs. BNTIC ne l’interprète pas comme une validation indépendante des résultats.