How many ring homomorphisms they are?
a) $\mathbb Z[x, \frac1x\ ] \to \mathbb Z_4$
b) $\mathbb Z[x]/(x^2) \to \mathbb Z_{24}$
c) $\mathbb Z[x]/(x^3-1) \to \mathbb C$
My try:
a) ?
b) Assume that exist homomorphism $\psi$.
$\mathbb Z[x] \xrightarrow{\pi} \mathbb Z [x]/(x^2) \xrightarrow{\psi} \mathbb Z_{24}$ transforms $x$ on such $y\in \mathbb Z_{24}$ that: $$x^2=0 \Rightarrow x=24k, k\in \left\{0,1,...\right\}$$
c) Assume that exist homomorphism $\psi$.
$\mathbb Z[x] \xrightarrow{\pi} \mathbb Z [x]/(x^3-1) \xrightarrow{\psi} \mathbb C$ transforms $x$ on such $y\in \mathbb C$ that: $$x^3-1=0 \Rightarrow x=1$$
However, I don't know what to do next
Assuming that ring homomorphisms send $1$ to $1$, it is enough to determine the possible images of $x$ or its class.
a) We need to send $x$ to a unit of $\mathbb Z_4$. The only possibilities are $1,3$.
b) We need to send $x$ to an element $y$ of $\mathbb Z_{24}$ such that $y^2=0$. The only possibilities are $0,12$.
b) We need to send $x$ to an element $z$ of $\mathbb C$ such that $z^3=1$. There are three possibilities.