Definition:Unit of Ring

Definition
A unit in a ring with unity $\left({R, +, \circ}\right)$ is an element of $R$ which is invertible under $\circ$ in $R$, i.e. that has an inverse.

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


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

Product Inverse
In a ring with unity $\left({R, +, \circ}\right)$, the inverse of $x \in U_R$ by $\circ$ is called the (ring) product inverse.

The usual means of denoting the product inverse of an element $x$ is by $x^{-1}$

Thus it is distinguished from the additive inverse of $x$, i.e. the negative of $x$, which is usually denoted $-x$.

Alternative Names
Some sources use the term invertible element.

Beware
Do not confuse a unit with the unity.