Irreducible element of a commutative monoid ($a=bc⟹b\in M^{\times }\text{or}c\in M^{\times }$)

A nonunit element of a commutative monoid is irreducible if each factorization has a unit factor. In a submonoid of the positive integers under multiplication, the only unit is one. Irreducibility is relative to the allowed factors: a composite integer can be irreducible in a smaller monoid even though it is not a prime number.