An element
of a ring which is nonzero, not a unit, and whose only divisors
are the trivial ones (i.e., the units and the products
, where
is a unit). Equivalently, an element
is irreducible if the only possible decompositions of
into the product of two factors are of
the form
where
is the multiplicative inverse of
.
The prime numbers and the irreducible polynomials are examples of irreducible elements. In a principal ideal domain, the irreducible elements are the generators of the nonzero prime ideals, hence the irreducible elements are exactly the prime elements. In general, however, the two notions are not equivalent.