Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1∈S and ab∈S if a,b∈S.
((a)) Define a relation ∼ on R×S by (a,s)∼(a′,s′) if there exists an s∗∈S such that s∗(s′a−sa′)=0. Show that ∼ is an equivalence relation on R×S.
((b)) Let a/s denote the equivalence class of (a,s)∈R×S and let S−1R be the set of all equivalence classes with respect to ∼. Define the operations of addition and multiplication on S−1R by
sa+tb=stat+bs,sa⋅tb=stab,
respectively. Prove that these operations are well-defined on S−1R and that S−1R is a ring with identity under these operations. The ring S−1R is called the ring of quotients of R with respect to S.
((c)) Show that the map ψ:R→S−1R defined by ψ(a)=a/1 is a ring homomorphism.
((d)) If R has no zero divisors and 0∈/S, show that ψ is one-to-one.
((e)) Prove that P is a prime ideal of R if and only if S=R∖P is a multiplicative subset of R.
((f)) If P is a prime ideal of R and S=R∖P, show that the ring of quotients S−1R has a unique maximal ideal. Any ring that has a unique maximal ideal is called a local ring.