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:

$n \cdot a = 0_R$

if and only if:

$p \divides n$

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 $24.8$