irreducible element

An irreducible element is one you cannot break apart into smaller honest pieces — the multiplicative atoms of a ring. It generalizes the idea of a prime number as something that admits only the trivial factorizations.

In an integral domain R, a nonzero nonunit element a is irreducible if whenever a = b·c with b and c in R, one of b or c must be a unit. In other words, every factorization of a is forced to be trivial, consisting of a times a unit. Units themselves and zero are excluded from the discussion by convention, so 'irreducible' is reserved for genuine building blocks.

Crucially, irreducible is not the same as prime. A prime element p satisfies the stronger property that p dividing a product b·c forces p to divide b or c. Every prime element is irreducible in a domain, but the converse can fail — in Z[√−5] the element 2 is irreducible yet not prime, since it divides 6 = (1 + √−5)(1 − √−5) without dividing either factor. The two notions coincide exactly in a unique factorization domain, and that coincidence is precisely what makes factorization unique.