Congruence submonoid of the positive integers ($S_m=\{n\ge 1:n\equiv 1(\mathrm{mod}m)\}$)

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.

The congruence submonoid of the positive integers with modulus ten has four distinct irreducible elements of a commutative monoid $21,221,51,91$ with equal pairwise products. Their ordinary prime factors are respectively $(3,7),(13,17),(3,17),(7,13)$. None of these proper prime factors is congruent to one modulo ten. The Euclid lemma proves that a product of two distinct primes has no other proper positive divisors, so all four are irreducible in this monoid. This gives nonunique factorization without assuming the fundamental theorem of arithmetic.