Unit (ring theory)

In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element $u$ of a ring $R$ is a unit if there exists $v$ in $R$ such that

vu=uv=1,

where $1$ is the multiplicative identity; the element $v$ is unique for this property and is called the multiplicative inverse of $u$ . The set of units of $R$ forms a group $R \times$ under multiplication, called the group of units or unit group of $R$ . Other notations for the unit group are $R *$ , $U(R)$ , and $E(R)$ (from the German term Einheit).

Less commonly, the term unit is sometimes used to refer to the element $1$ of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, $1$ 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.

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