Integers under Multiplication form Countably Infinite Commutative Monoid

Theorem

The set of integers under multiplication $\struct {\Z, \times}$ is a countably infinite commutative monoid.

Proof

First we note that Integers under Multiplication form Monoid.

$\Box$

Then we have:

Commutativity

Integer Multiplication is Commutative.

$\Box$

Infinite

Integers are Countably Infinite.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.2$
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $2$: Examples of Groups and Homomorphisms: $14$