Integers form Subring of Reals

Theorem

The ring of integers $\struct {\Z, +, \times}$ forms a subring of the field of real numbers.

Proof

We have that the set of integers $\Z$ are a subset of the real numbers $\R$.

The field of real numbers is, a fortiori, also a ring.

Hence the result, by definition of subring.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 56$. Subrings and Subfields
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subring