Unit group
In mathematics, an invertible element or a unit in a (unital) ring 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 opposite −1R are always units. Hence, pairs of additive inverse elementsx and −x are always associated.
In any ring, 1 is a unit. More generally, any root of unity in a ring R is a unit: if rn = 1, then rn − 1 is a multiplicative inverse of r. On the other hand, 0 is never a unit. A ring R is a field (possibly non-commutative, also known as a skew field or division ring) if and only if U(R) = R ∖ {0}. For example, the units of the real numbers R are R ∖ {0}. Thus, for any ring R, there is an inclusion
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