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