Unital rings, homomorphisms, etc

[calvino]

I've already completed 1), but it's necessary for one to know it for question 2). I'm pretty sure that I've found my homomorphism in 2, but I don't know whether or not is unique. How do I show a homomorphism is unique in this case?

Problem 1: Let R be a commutative unital ring, and let S be a multiplicative submonoid of R. Define an equivalence relation ~ on R x S by (a,s)~(b,t) if there is r in S with rat = rbs. Let a/s denote the ~-equivalence class of (a,s). Show that with

a/s + b/t = (at+bs)/st and (a/s)(b/t) = ab/st

one can make RxS/~ into a commutative, unital ring, and that j(a) = a/1 defines a homomorphism j of unital rings from R into RxS/~ that maps S into the group of invertible elements of RxS/~

Problem 2 (=continuation of Problem 1): Let R and S be as above, and let phi: R --> T be a homomorphism that maps S into the group of invertible elements of the commutative unital ring T. Show that there is unique homomorphism psi: RxS/~ ---> T of unital rings with psi.j = phi.

 

[StatusX]

The condition psi(j(r))=phi(r) tells you that psi(r/1)=phi(r). What must psi(1/s) be?

 

[calvino]

i understand how to define the function (i think). Am i suppose to see it's uniqueness, naturally?

 

[StatusX]

How did you find the homomorphism? What steps did you take? At each step, can you argue that there is no other choice you could have made that would still leave you with a function satisfying the necessary conditions? If so, you have uniqueness.

 

[calvino]

can we use the fundamental theorm of homomorphisms?

i didn't. i simple constructed a diagram of the homomorphisms involved and showed that since psi.j is in a sense doing the same things as phi is, then they method is unique.

 

[StatusX]

I have no idea what "since psi.j is in a sense doing the same things as phi is, then they method is unique" means. Like I was saying before, psi(r/1)=phi(r). Can you show you only have one choice for psi(1/s)? Then psi(r/s)=psi(r/1)psi(1/s) is unique.