Invertible
Concept history
A revision trail for this concept page.
Revision 784
8/1/2026, 9:38:34 AM · Sequoia
Concept edited
Recorded titleInvertible
Recorded typeDefinition
Compare with revision 7834 changed lines
1
An element $x$ from a [[unital|unital]] [[magma|magma]] $(M,*)$, with [[Identity element|identity]] $e$, is said to be invertible if there exist $y,z\in M$ such that $x*y=e$ and $z*x=e$.2
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##### Remarks4
- If the operation $*$ is [[monoid|associative]], hence if $(M,*)$ is a [[monoid|monoid]], then one can set $y=z$ and say that $y$ is *the* inverse of $x$. It is necessarily unique in this case.4
- If the operation $*$ is [[monoid|associative]], hence if $(M,*)$ is a [[monoid|monoid]], then one can set $y=z$ and say that $y$ is *the* inverse of $x$, sometimes denoted as $x^{*-1}$. It is necessarily unique in this case.5
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- In the case of a ring $(R,+,\times)$, then $(R,\times)$ is a [[monoid|monoid]] hence each invertible element has to be unique. We denote by $x^{-1}$ the inverse of $x$ if it exists.Revision 783
8/1/2026, 9:35:13 AM · Sequoia
Concept edited
This older revision predates detailed metadata tracking.
Revision 778
8/1/2026, 9:27:49 AM · Sequoia
Concept edited
Compare with revision 7582 changed lines
1
An element $x$ from a [[unital|unital]] [[magma|magma]] $(M,*)$, with [[Identity element|identity]] $e$, is said to be invertible if there exist $y,z\in M$ such that $x*y=e$ and $z*x=e$.2
3
##### Remarks4
- If the operation $*$ is [[associative|associative]], hence if $(M,*)$ is a [[monoid|monoid]], then one can set $y=z$ and say that $y$ is *the* inverse of $x$. It is necessarily unique in this case.4
- If the operation $*$ is [[monoid|associative]], hence if $(M,*)$ is a [[monoid|monoid]], then one can set $y=z$ and say that $y$ is *the* inverse of $x$. It is necessarily unique in this case.Revision 758
7/31/2026, 1:25:59 PM · Ancient Tree
Concept edited
Compare with revision 7515 changed lines
1
An element $x$ from a [[unital|unital]] [[magma|magma]] $(M,*)$, with [[neutral|neutral]] $e$, is said to be invertible if there exist $y,z\in M$ such that $x*y=e$ and $z*x=e$.1
An element $x$ from a [[unital|unital]] [[magma|magma]] $(M,*)$, with [[Identity element|identity]] $e$, is said to be invertible if there exist $y,z\in M$ such that $x*y=e$ and $z*x=e$.2
3
If the law $*$ is [[associative|associative]], hence if $(M,*)$ is a [[monoid|monoid]], then one can set $y=z$ and say that $y$ is the inverse of $x$. It is necessarily unique in this case.3
##### Remarks4
- If the operation $*$ is [[associative|associative]], hence if $(M,*)$ is a [[monoid|monoid]], then one can set $y=z$ and say that $y$ is *the* inverse of $x$. It is necessarily unique in this case.Revision 751
7/31/2026, 10:05:41 AM · Sequoia
Concept edited
This older revision predates detailed metadata tracking.
Revision 744
7/31/2026, 9:47:12 AM · Sequoia
Concept edited
This older revision predates detailed metadata tracking.
Revision 743
7/31/2026, 9:47:02 AM · Sequoia
Concept created
An element $x$ from a [[unital|unital]] [[magma|magma]] $(M,*)$, with [[neutral|neutral]] $e$, is said to be invertible if there exist $y,z\in M$ such that $x*y=e$ and $z*x=e$. If the law $*$ is [[associative|associative]], hence if $(M,*)$ is a [[monoid|monoid]], then one can set $y=z$ and say that $y$ is the inverse of $x$. It is necessarily unique in this case.