Nonunit

Nonunit is a term used in ring theory to refer to an element that is not a unit. In a ring R with a multiplicative identity, a unit is an element that has a multiplicative inverse in R; equivalently, an element a is a unit if there exists b in R such that ab = ba = 1. The set of all units of R is denoted U(R) and forms a group under multiplication. The nonunits are the elements of R that are not invertible.

Examples illustrate the concept. In the ring of integers Z, the only units are 1 and -1, so every other integer is a nonunit. In the polynomial ring F[x] over a field F, the units are the nonzero constant polynomials; polynomials of positive degree are nonunits. In the matrix ring M_n(F) over a field F, the units are the invertible matrices (those with nonzero determinant); singular matrices are nonunits.

Several properties help distinguish units from nonunits. The zero element is always a nonunit in a ring with identity. Nonunits often include zero divisors and nilpotent elements, though the precise composition depends on the ring. In a finite ring with unity, every element is either a unit or a zero divisor. In a commutative ring, an element is a unit if and only if it is not contained in any maximal ideal; equivalently, nonunits lie in at least one maximal ideal.

The notion of nonunit is used in various areas of algebra, including the study of factorization, localization, and the structure of rings. Identifying units versus nonunits helps organize elements by their invertibility and informs constructions that require inverting certain elements.