# Integers under Multiplication form Monoid

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## Theorem

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

## Proof

From Integers under Multiplication form Semigroup, $\struct {\Z, \times}$ is a semigroup.

From Integer Multiplication Identity is One, $\struct {\Z, \times}$ has an identity element, which is $1$.

All the criteria for being a monoid are thus seen to be fulfilled.

$\blacksquare$

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids