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 it 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.