Definition:Unit of Ring

Definition
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.

Then $x \in R$ is a unit in $\struct {R, +, \circ}$ $x$ is invertible under $\circ$.

That is, a unit of $R$ is an element of $R$ which has an inverse.

The group of units of a ring $\struct {R, +, \circ}$ is often denoted $U_R$ (or just $U$ if there is no doubt what the ring is that's being talked about).

Thus:


 * $x \in U_R \iff \exists y \in R: x \circ y = 1_R = y \circ x$

Also known as
Some sources use the term invertible element.

The notations $R^*$ and $R^\times$ are also seen for the set of units $U_R$.

Beware
Do not confuse a unit with the unity.