Subrings of Integers are Sets of Integer Multiples/Examples/Even Integers

Theorem

Let $2 \Z$ be the set of even integers.

Then $\struct {2 \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$.

Proof

From Subrings of Integers are Sets of Integer Multiples, a ring of the form $\struct {n \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$ when $n \ge 1$.

$\struct {2 \Z, +, \times}$ is such an example.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 19$. Subrings: Example $31$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subring