An element a of a ring which is nonzero, not a unit, and whose only divisors are the trivial ones (i.e., the units and the products u a, where u is a unit). Equivalently, an element a is irreducible if the only possible decompositions of a into the product of two factors are of the form a = u^(-1)·u a, where u^(-1) is the multiplicative inverse of u.