# 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 $p$.

Let $g_a: \Z \to R$ be the mapping from the integers into $R$ defined as:

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

where $\cdot$ denotes the multiple operation.

Then the following hold:

### Multiple Function on Ring is Homomorphism

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

### Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function

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

where:

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

### Multiplication Function on Ring with Unity is Zero if Characteristic is Divisor

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

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

### Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic

Let $a \in R$ such that $a$ is not a zero divisor of $R$.

Then:

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

where:

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

### Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic

Let $a \in R$ such that $a$ is not a zero divisor of $R$.

Then:

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

where:

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

### Kernel of Multiple Function on Ring with Characteristic Zero is Trivial

Let $a \in R$ such that $a$ is not a zero divisor of $R$.

Let the characteristic of $R$ be $0$.

Then:

- $\map \ker {g_a} = \set {0_R}$

where $\ker$ denotes the kernel of $g_a$.

### Multiple Function on Ring is Zero iff Characteristic is Divisor

Let $a \in R$ such that $a$ is not a zero divisor of $R$.

Then:

- $n \cdot a = 0_R$

- $p \divides n$

## Examples

### Characteristic $2$

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 $2$.

Then:

- $\forall a \in R: 2 \cdot a = 0$

or equivalently:

- $\forall a \in R: a = -a$

## Sources

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