Let R be a commutative ring with identity, I an ideal of R and : R

Let R be a commutative ring with identity, I an ideal of R and π : R → R/ I the canonical projection.

(a) If S is a multiplicative subset of R, then πS = π(S) is a multiplicative subset of R/ I.

(b) The mapping θ: s^-1R → (πS)^-1(R/I) given by r/s|→ π(r)/π(s) is a well defined function.

(c) θ is a ring epimorphism with kernel s^-1I and hence induces a ring isomorphism s^-1R/s^-1I ≅ (πS)^-1(R/I)