Morphism from the Gaussian integers

Let i be the imaginary number, let Z[i]={a+bia,bZ}, a principal ideal domain, and let Z2 be the finite ring of integers modulo 2.

  1. Define a ring homomorphism from Z[i]Z2. You must prove it is a ring homomorphism.
  2. Find, with proof, a generator for the kernel of your ring homomorphism.