Definition:Unit of Ring

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Not to be confused with Definition:Unity of Ring.

Definition

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 $1_R$.


Product Inverse

The inverse of $x \in U_R$ by $\circ$ is called the (ring) product inverse of $x$.


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 $x$, that is, the (ring) negative of $x$, which is usually denoted $-x$.


Group of Units

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


Then the set $U_R$ of units of $\struct {R, +, \circ}$ is called the group of units of $\struct {R, +, \circ}$.


This can be denoted explicitly as $\struct {U_R, \circ}$.


Also known as

Some sources use the term invertible element for unit of ring.


Also see

  • Results about units of rings can be found here.


Sources