Ring Homomorphisms

Ring Homomorphisms

A ring homomorphism $\phi: R \to S$ is a function that preserves both operations:

$\phi(a + b) = \phi(a) + \phi(b)$ $\phi(a \cdot b) = \phi(a) \cdot \phi(b)$

If $R$ and $S$ have unity, we also require $\phi(1_R) = 1_S$.

The Natural Projection

The map $\phi: \mathbb{Z}_m \to \mathbb{Z}_n$ defined by $\phi(a) = a \bmod n$ is a ring homomorphism whenever $n$ divides $m$.

For example, $\phi: \mathbb{Z}_{12} \to \mathbb{Z}_4$ with $\phi(a) = a \bmod 4$:

• $\phi(7 + 8) = \phi(3) = 3$ and $\phi(7) + \phi(8) = 3 + 0 = 3$ in $\mathbb{Z}_4$ ✓
• $\phi(3 \cdot 5) = \phi(3) = 3$ and $\phi(3) \cdot \phi(5) = 3 \cdot 1 = 3$ in $\mathbb{Z}_4$ ✓

Kernel of a Ring Homomorphism

$\ker(\phi) = \{a \in R : \phi(a) = 0_S\}$

The kernel is always an ideal of $R$. For $\phi: \mathbb{Z}_{12} \to \mathbb{Z}_4$, the kernel is $\{0, 4, 8\}$.

Image

$\text{im}(\phi) = \{\phi(a) : a \in R\}$

The image is a subring of $S$.

Your Task

Implement is_ring_hom(phi, m, n) that checks if a function (given as a list) is a ring homomorphism from $\mathbb{Z}_m \to \mathbb{Z}_n$, ring_kernel(phi, m) for the kernel, and ring_image(phi, m) for the image.