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.