In mathematics, an invertible element or a unit in a ring with identity R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such that

The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring.

The term unit is also used to refer to the identity element 1R of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. For this reason, some authors call 1R "unity" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

The multiplicative identity 1R and its additive inverse −1R are always units. Hence, pairs of additive inverse elements^[3] x and −x are always associated.

1 is a unit in any ring. More generally, any root of unity in a ring R is a unit: if r**n = 1, then r**n − 1 is a multiplicative inverse of r. On the other hand, 0 is never a unit (except in the zero ring). A ring R is called a skew-field (or a division ring) if U(R) = R - {0}, where U(R) is the group of units of R (see below). A commutative skew-field is called a field. For example, the units of the real numbers R are R - {0}.

In the ring of integers Z, the only units are +1 and −1.

in the ring Z[1 + √5/2], and in fact the unit group of this ring is infinite.

In fact, Dirichlet's unit theorem describes the structure of U(R) precisely: it is isomorphic to a group of the form

In the ring Z/nZ of integers modulo n, the units are the congruence classes (mod n) represented by integers coprime to n. They constitute the multiplicative group of integers modulo n.

For a commutative ring R, the units of the polynomial ring R[x] are precisely those polynomials

The unit group of the ring Mn(R) of n × n matrices over a commutative ring R (for example, a field) is the group GLn(R) of invertible matrices.

See also Hua's identity for a similar type of results.

The units of a ring R form a group U(R) under multiplication, the group of units of R. Other common notations for U(R) are R∗, R×, and E(R) (from the German term Einheit).

A commutative ring is a local ring if R − U(R) is a maximal ideal. As it turns out, if R − U(R) is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from U(R).

The formulation of the group of units defines a functor U from the category of rings to the category of groups: every ring homomorphism f : R → S induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units. This functor has a left adjoint which is the integral group ring construction.

In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ∼ on R called associatedness such that

means that there is a unit u with r = us.

In an integral domain the cardinality of an equivalence class of associates is the same as that of U(R).