 Congruence submonoid of the positive integers

For an integer $m\ge 2$, 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.