Homomorphism from Integers into Ring with Unity

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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.

By General Distributivity Theorem, 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 $n \cdot 1 = 0$ we have $n \cdot a = 0$.



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