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 $24.8$