Existence of certain ring morphisms

#ring_theory

Let $Z [i] = {a + b i ∣ a, b \in Z, i^{2} = - 1}$ , a subring of $C$ . Prove there is no ring homomorphism $Z [i] \to Z_{19}$ , but there is a ring homomorphism $Z [i] \to Z_{13}$ . Note a ring homomorphism of commutative rings with $1$ must send $1$ to $1$ .

Hint: The group of units in $Z_{19}$ is the cyclic group $U (19)$ of order 18, and the group of units in $Z_{13}$ is the cyclic group $U (13)$ of order 12.

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Let ${\bf Z}[i]=\{a+bi\,\mid \, a,b\in {\bf Z},\, i^2=-1\}$, a subring of ${\bf C}$. Prove there is no ring homomorphism ${\bf Z}[i]\to {\bf Z}_{19}$, but there is a ring homomorphism ${\bf Z}[i]\to {\bf Z}_{13}$. Note a ring homomorphism of commutative rings with $1$ must send $1$ to $1$.

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\noindent {\itshape Hint:} The group of units in ${\bf Z}_{19}$ is the cyclic group $U(19)$ of order 18, and the group of units in ${\bf Z}_{13}$ is the cyclic group $U(13)$ of order 12.