An irreducible element 
of a ring is nonzero, is not a unit,
and has only the trivial divisors, namely the units and the products 
, where 
is a unit. Equivalently, an element 
is irreducible if its only decompositions 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.
