# Homomorphism from Integers into Ring with Unity

## Theorem

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let the characteristic of $R$ be $p$.

For any $a \in R$, we define the mapping $g_a: \Z \to R$ from the integers into $R$ as:

- $\forall n \in \Z: \map {g_a} n = n \cdot a$

Then $g_a$ is a group homomorphism from $\struct {\Z, +}$ to $\struct {R, +}$.

Also:

- $\ideal p \subseteq \map \ker {g_a}$

where:

- $\map \ker {g_a}$ is the kernel of $g_a$;
- $\ideal p$ is the principal ideal of $\Z$ generated by $p$.

Also:

- $p \divides n \implies n \cdot a = 0$

where $p \divides n$ denotes that $p$ is a divisor of $n$.

## Proof

The fact that $g_a$ is a group homomorphism follows directly from Index Laws for Monoids:

\(\ds \map {g_a} m + \map {g_a} n\) | \(=\) | \(\ds m \cdot a + n \cdot a\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {m + n} \cdot a\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map {g_a} {m + n}\) |

By Multiple of Ring Product, we have that:

- $\forall n \in \Z_{>0}: \paren {n \cdot x} \circ y = n \cdot \paren {x \circ y} = x \circ \paren {n \cdot y}$

So:

- $\forall n \in \Z_{>0}: n \cdot a = \paren {n \cdot a} \circ 1_R = a \circ \paren {n \cdot 1_R}$

so when $n \cdot 1_R = 0$ we have $n \cdot a = 0$.

For $n \in \Z_{<0}$, we have $-n \in \Z_{>0}$.

So:

- $n \cdot a = -\paren {-n} \cdot a = -\paren {\paren {-n \cdot a} \circ 1_R} = -\paren {a \circ \paren {-n \cdot 1_R}}$

so when $-n \cdot 1_R = 0$ we have $n \cdot a = 0$.

For $n = 0$, we trivially have $n \cdot a = 0$.

By definition of characteristic, we have:

- $p \divides n \iff n \cdot 1_R = 0$

by the above, this implies $n \cdot a = 0$.

Therefore:

- $p \divides n \implies n \cdot a = 0$

We also have:

\(\ds n\) | \(\in\) | \(\ds \ideal p\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds p\) | \(\divides\) | \(\ds n\) | Definition of Ideal of Ring | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds n \cdot a\) | \(=\) | \(\ds 0\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds n\) | \(\in\) | \(\ds \map \ker {g_a}\) | Definition of Kernel of Group Homomorphism |

By definition of subset:

- $\ideal p \subseteq \map \ker {g_a}$

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.8 \ 1^\circ$