Integral Multiple Distributes over Ring Addition
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Theorem
Let $\struct {R, +, \times}$ be a ring, or a field.
Let $a, b \in R$ and $m, n \in \Z$.
Then:
- $(1): \quad \paren {m + n} \cdot a = \paren {m \cdot a} + \paren {n \cdot a}$
- $(2): \quad m \cdot \paren {a + b} = \paren {m \cdot a} + \paren {m \cdot b}$
where $m \cdot a$ is as defined in integral multiple.
Proof
We have that the additive group $\struct {R, +}$ is an abelian group.
$(1): \quad \paren {m + n} \cdot a = \paren {m \cdot a} + \paren {n \cdot a}$:
This is an instance of Powers of Group Elements: Sum of Indices when expressed in additive notation:
- $\forall n, m \in \Z: \forall a \in R: m a + n a = \paren {m + n} a$
$\Box$
$(2): \quad m \cdot \paren {a + b} = \paren {m \cdot a} + \paren {m \cdot b}$:
This is an instance of Power of Product in Abelian Group when expressed in additive notation:
- $\forall n \in \Z: \forall a, b \in R: n \paren {a + b} = n a + n b$
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties