Real Numbers form Ring
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Theorem
The set of real numbers $\R$ forms a ring under addition and multiplication: $\struct {\R, +, \times}$.
Proof
From Real Numbers under Addition form Infinite Abelian Group, $\struct {\R, +}$ is an abelian group.
We also have that:
- Real Multiplication is Closed:
- $\forall x, y \in \R: x \times y \in \R$
- Real Multiplication is Associative:
- $\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$
Thus $\struct {\R, +}$ is a semigroup.
Finally we have that Real Multiplication Distributes over Addition:
\(\ds \forall x, y, z \in \R: \, \) | \(\ds x \times \paren {y + z}\) | \(=\) | \(\ds x \times y + x \times z\) | |||||||||||
\(\ds \paren {y + z} \times x\) | \(=\) | \(\ds y \times x + z \times x\) |
Hence the result, by definition of ring.
$\blacksquare$
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $4$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 54$. The definition of a ring and its elementary consequences