Unit (ring theory)

In algebra, a unit (or invertible element) in a ring is something that can be multiplied by another element to give 1, the ring’s multiplicative identity.

If an element u in a ring R has another element v in R such that:

uv=vu=1

then u is a unit, and v is its multiplicative inverse. All the units in R together form a group under multiplication, called the group of units, written as R×, R∗, U(R), or E(R).

Sometimes the word unit just means the identity element 1, but to avoid confusion, “rings with identity” or “rings with unity” are usually specified.

Examples

• In the integers Z, the only units are 1 and -1.
• In Z/nZ (integers mod n), a number is a unit if it is coprime to n. These units form the group (Z/nZ)×.
• In Z[3], 2+3 is a unit because (2+3)(2−3)=1. This ring has infinitely many units.
• In a division ring, every nonzero element is a unit. If the ring is also commutative, it’s a field.

Units in Other Rings

• Polynomial ring R[x]: A polynomial a0+a1x+⋯+anxn is a unit if a0 is a unit and all other ai are nilpotent (some power equals 0). If there are no nilpotents (for example, R is a field), then only constant polynomials that are units in R count.
• Power series ring R[[x]]: A series is a unit if its constant term a0 is a unit.
• Matrix ring Mn(R): A matrix is a unit (invertible) if its determinant is a unit in R. The group of all invertible n×n matrices is called GLn(R).

Group of Units and Local Rings

A local ring is one where the set of non-units forms a single maximal ideal.

In a finite field R, the unit group R× is always cyclic with order ∣R∣−1.

Every ring homomorphism maps units to units, so taking the group of units is a functor from rings to groups.

Associated Elements

In a commutative ring, two elements r and s are associates if r=us for some unit u.

This forms an equivalence relation, meaning the ring can be partitioned into classes of elements that differ only by multiplication by units.

Example: In Z, 6 and -6 are associates because -1 is a unit.