Nonunits

Nonunits are elements of a ring with identity that do not have a multiplicative inverse. An element u is a unit if there exists v such that uv = vu = 1. The set of all units forms a group under multiplication, called the unit group U(R). The complement R \ U(R) is the set of nonunits; it contains 0 and any noninvertible elements.

Examples illustrate the idea. In the ring of integers Z, the only units are 1 and -1, so every other integer is a nonunit. In the polynomial ring Z[x], the units are again ±1, so all other polynomials are nonunits. In the ring of continuous functions C(X) on a space X, a function is a unit precisely when it is nowhere zero; functions that vanish somewhere are nonunits.

Key properties are useful for intuition. If a is a nonunit and b is a unit, then the product ab is a nonunit; multiplying by a unit cannot turn a nonunit into a unit. The set of nonunits is not generally an ideal. In a commutative ring with identity, every nonunit lies in some maximal ideal; equivalently, a ring is local if and only if the nonunits form the unique maximal ideal.

The concept helps organize factorization and localization ideas in algebra. In a field, every nonzero element is a unit, so the only nonunit is zero. In more general rings, nonunits capture where invertibility fails and interact with ideals and the structure of the ring’s unit group.