- #1

calvino

- 108

- 0

**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.

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.