Unit (ring theory)

where 1 is the multiplicative identity.[1][2] The set of units U(R) of a ring forms a group under multiplication.

Less commonly, the term unit is also used to refer to the element 1 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 1 "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 ring of integers in a number field may have more units in general. For example, in the ring Z[1 + 5/ 2] that arises by adjoining the quadratic integer 1 + 5/ 2 to Z, one has

in the ring, so 5 + 2 is a unit. (In fact, the unit group of this ring is infinite.[citation needed])

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 ring R is the group GLn(R) of invertible matrices. For a commutative ring R, an element A of Mn(R) is invertible if and only if the determinant of A is invertible in R. In that case, A−1 is explicitly given by Cramer's rule.

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).

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 : RS 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.[6]

Suppose that R is commutative. Elements r and s of R are called associate if there exists a unit u in R such that r = us; then write rs. In any ring, pairs of additive inverse elements[a] x and x are associate. For example, 6 and −6 are associate in Z. In general, ~ is an equivalence relation on R.

Associatedness can also be described in terms of the action of U(R) on R via multiplication: Two elements of R are associate if they are in the same U(R)-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as U(R).

The equivalence relation ~ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring R.