Let $a+bi\in\mathbb{Z}[i]$ with $\gcd(a,b)=1$.
I know that $\mathbb{Z}[i]/\langle a+bi\rangle\cong\mathbb{Z}_{a^{2}+b^{2}}$ by a ring homomorphism $\phi:\mathbb{Z}[i]\to\mathbb{Z}_{a^{2}+b^{2}}$, which maps $x+yi$ to $x-(ab^{-1})y\pmod{a^{2}+b^{2}}$ with $\ker\phi=\langle a+bi\rangle$.
My question is:
Is there a ring homomorphism as above $\phi$ if $\gcd(a,b)=d>1$?
I found the relation $\mathbb{Z}[i]/\langle a+bi\rangle\cong\mathbb{Z}_{d}\times\mathbb{Z}_{(a^{2}+b^{2})/d}$ if $\gcd(a,b)=d>1$.
But, i couldn't find the ring homomorphism between $\mathbb{Z}[i]$ and $\mathbb{Z}_{d}\times\mathbb{Z}_{(a^{2}+b^{2})/d}$ with kernel $\langle a+bi\rangle$
Give some advice! Thank you!