Category:Definitions/Units of Rings

This category contains definitions related to Units of Rings.
Related results can be found in Category:Units of Rings.

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

Definition 1

An element $x \in R$ is a unit of $\struct {R, +, \circ}$ if and only if $x$ is invertible under $\circ$.

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

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

Definition 2

An element $x \in R$ is a unit of $\struct {R, +, \circ}$ if and only if $x$ is divisor of $1_R$.