Congruence submonoid of the positive integers
= Congruence submonoid of the positive integers
{title2=$S_m=\{n\geq1:n\equiv1\pmod m\}$}
For an integer $m\geq2$, the positive integers congruent to one modulo $m$ form a <cancellative commutative monoid> under multiplication. The identity is one, and it is the only unit. Proper factors outside this congruence class do not witness reducibility inside the monoid. Hence <irreducible elements of a commutative monoid> need not coincide with ordinary <prime numbers>.