Characteristic of a ring

Let R be a commutative ring with 1. The characteristic char(R) of R is the unique integer n0 such that nZ is the kernel of the homomorphism θ:ZR given by

θ(m)={1R++1Rm, if m01R++1R|m|, if m<0
  1. Prove that if f:RS is a monomorphism of commutative rings with 1, then char(R)=char(S).
  2. Prove by given an example that char(R) is not always preserved by ring homomorphisms.