1. **Problem statement:** Find all ring homomorphisms from $\mathbb{Z}_6$ to $\mathbb{Z}_6$ and from $\mathbb{Z}_{20}$ to $\mathbb{Z}_{30}$. 2. **Recall:** A ring homomorphism $\varphi: \mathbb{Z}_n \to \mathbb{Z}_m$ is determined by $\varphi(1)$ because $\varphi(k) = k \cdot \varphi(1)$ for all $k \in \mathbb{Z}_n$. 3. **Condition for ring homomorphism:** Since $\varphi$ respects multiplication, we must have $\varphi(1)^2 = \varphi(1)$ in $\mathbb{Z}_m$, i.e., $\varphi(1)$ is an idempotent element in $\mathbb{Z}_m$. Also, $\varphi(n) = \varphi(0) = 0$ implies $n \cdot \varphi(1) = 0$ in $\mathbb{Z}_m$. --- ### Part 1: Homomorphisms from $\mathbb{Z}_6$ to $\mathbb{Z}_6$ 4. We want $\varphi(1) = e \in \mathbb{Z}_6$ such that: - $e^2 = e \pmod{6}$ (idempotent) - $6e = 0 \pmod{6}$ (automatically true since $6e$ is multiple of 6) 5. Check idempotents in $\mathbb{Z}_6$: - $0^2 = 0$ - $1^2 = 1$ - $2^2 = 4 \neq 2$ - $3^2 = 9 \equiv 3$ - $4^2 = 16 \equiv 4$ - $5^2 = 25 \equiv 1 \neq 5$ 6. Idempotents are $0,1,3,4$. So possible $\varphi(1)$ are these. 7. Each $\varphi$ is defined by $\varphi(k) = k \cdot e \pmod{6}$ for $e \in \{0,1,3,4\}$. --- ### Part 2: Homomorphisms from $\mathbb{Z}_{20}$ to $\mathbb{Z}_{30}$ 8. We want $\varphi(1) = e \in \mathbb{Z}_{30}$ such that: - $e^2 = e \pmod{30}$ (idempotent) - $20e = 0 \pmod{30}$ 9. First find idempotents in $\mathbb{Z}_{30}$. Since $30 = 2 \cdot 3 \cdot 5$, use Chinese remainder theorem: - $e \equiv e^2 \pmod{2}$ implies $e \equiv 0$ or $1 \pmod{2}$ - $e \equiv e^2 \pmod{3}$ implies $e \equiv 0$ or $1 \pmod{3}$ - $e \equiv e^2 \pmod{5}$ implies $e \equiv 0$ or $1 \pmod{5}$ 10. So $e$ is congruent to either 0 or 1 modulo each prime factor. Possible combinations: - $(0,0,0)$ - $(0,0,1)$ - $(0,1,0)$ - $(0,1,1)$ - $(1,0,0)$ - $(1,0,1)$ - $(1,1,0)$ - $(1,1,1)$ 11. Use CRT to find $e$ modulo 30 for each combination: - $(0,0,0) \to 0$ - $(0,0,1) \to 25$ - $(0,1,0) \to 15$ - $(0,1,1) \to 10$ - $(1,0,0) \to 6$ - $(1,0,1) \to 1$ - $(1,1,0) \to 21$ - $(1,1,1) \to 16$ 12. Now check $20e \equiv 0 \pmod{30}$ for each $e$: - $20 \cdot 0 = 0$ mod 30 ✓ - $20 \cdot 25 = 500 \equiv 20$ mod 30 ✗ - $20 \cdot 15 = 300 \equiv 0$ mod 30 ✓ - $20 \cdot 10 = 200 \equiv 20$ mod 30 ✗ - $20 \cdot 6 = 120 \equiv 0$ mod 30 ✓ - $20 \cdot 1 = 20$ mod 30 ✗ - $20 \cdot 21 = 420 \equiv 0$ mod 30 ✓ - $20 \cdot 16 = 320 \equiv 20$ mod 30 ✗ 13. Valid $e$ are $0, 15, 6, 21$. 14. Each homomorphism is $\varphi(k) = k \cdot e \pmod{30}$ for $e \in \{0,6,15,21\}$. --- **Final answers:** - Homomorphisms $\mathbb{Z}_6 \to \mathbb{Z}_6$ correspond to $e \in \{0,1,3,4\}$. - Homomorphisms $\mathbb{Z}_{20} \to \mathbb{Z}_{30}$ correspond to $e \in \{0,6,15,21\}$.