Finding homomorphisms and kernels from a given ring R

Give the following rings $R$ and ideals $I$ find a ring $S$ and a homomorphism $f:R \rightarrow S$ with kernel $I$

i) $R=\mathbb{Q} [x], I=(x^{2}-2)R$

ii)$R=\mathbb{Z}[i], I=2R$ (Gaussian Integers)

With both I'm struggling to find a homomorphism which satisfies the kernel. I would assume for the Gaussian Integers the norm function could be applied?

Any help would be great :)

• Doesn't $\nu:R\mapsto R/I$ with $r\mapsto r+I$ count? – Pedro Tamaroff Nov 1 '13 at 20:16

For the first one, consider the evaluation homomorphism $$\xi:\Bbb Q[x]\to\Bbb Q[\sqrt 2]$$ wth $P\mapsto P(\sqrt 2)$. Then $\xi P=\xi Q$ iff $(P-Q)(\sqrt 2)=0$ iff $x^2-2\mid P-Q$ iff $P-Q\in (x^2-2)$.
so $\ker \xi =(x^2-2)$.
For the second one, consider $\eta:\Bbb Z[i]\mapsto (\Bbb Z/(2))[i]$ where $P$'s coefficients are mapped to their class modulo 2. Then $P\mapsto 0$ iff all its coefficients are divisible by two, iff $P\in (2\Bbb Z)[i]$.