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# left and right unity of ring

If a ring $(R,\,+,\,\cdot)$ contains a multiplicative left identity element $e$, i.e. if

$e\cdot a=a\quad\forall a,$ |

then $e$ is called the left unity of $R$.

If a ring $R$ contains a multiplicative right identity element $e^{{\prime}}$, i.e. if

$a\cdot e^{{\prime}}=a\quad\forall a,$ |

then $e^{{\prime}}$ is called the right unity of $R$.

A ring may have several left or right unities (see e.g. the Klein four-ring).

If a ring $R$ has both a left unity $e$ and a right unity $e^{{\prime}}$, then they must coincide, since

$e^{{\prime}}=e\cdot e^{{\prime}}=e.$ |

This situation means that every right unity equals to $e$, likewise every left unity. Then we speak simply of a unity of the ring.

Defines:

left unity, right unity

Related:

InversesInRings

Major Section:

Reference

Type of Math Object:

Definition

Parent:

## Mathematics Subject Classification

20-00*no label found*16-00

*no label found*

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