unit

In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term unit is sometimes used to refer to the element of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More generally, any root of unity in a ring is a unit: if , then is a multiplicative inverse of . In a nonzero ring, the element 0 is not a unit, so is not closed under addition. A nonzero ring in which every nonzero element is a unit (that is, ) is called a division ring (or a skew-field). A commutative division ring is called a field. For example, the unit group of the field of real numbers is .