Inverse element

Builds on Binary operation Identity element

Definition

Let $S$ be a set with a binary operation $\ast$ and an identity element $e$ . Let $a, b \in S$ . The element $b$ is a left inverse of $a$ when $b \ast a = e$ . The element $b$ is a right inverse of $a$ when $a \ast b = e$ . The element $b$ is an inverse of $a$ (or two-sided inverse) when it is both a left inverse and a right inverse of $a$ . An element that has an inverse is called invertible.

Remarks

For addition on $\mathbb{Z}$ , the inverse of $a$ is $-a$ . For multiplication on $\mathbb{Q}$ , each nonzero $a$ has the inverse $1/a$ , and $0$ has no inverse. When $\ast$ is associative, an element has at most one inverse.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Binary operation definition
Identity element definition
Inverse element definition

Where this leads

23 pages build on this one.

Tag	`0BRHKKP333`
URI	`m2://0BRHKKP333`
Kind	definition
Writing	W1 stated
MSC	08A02
Defines	Inverse element
Uses	Binary operation, Identity element

Inverse element

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by