Multiple Function on Ring is Homomorphism

Theorem

Let $\struct {R, +, \circ}$ be a ring.

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:

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

Proof

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

\(=\)

\(\ds m \cdot a + n \cdot a\)

Definition of Integral Multiple

\(\ds \)

\(=\)

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

Integral Multiple Distributes over Ring Addition

\(\ds \)

\(=\)

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

Definition of Integral Multiple

$\blacksquare$

Sources

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