Rational Numbers form Ring

Theorem

The set of rational numbers $\Q$ forms a ring under addition and multiplication: $\struct {\Q, +, \times}$.


Proof

Recall that $\struct {\Q, +, \times}$ is a field.

As a field is also by definition a division ring, which is an example of a ring, the result follows.

$\blacksquare$


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 $4$
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