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# unity

The unity of a ring $(R,\,+,\,\cdot)$ is the multiplicative identity of the ring, if it has such. The unity is often denoted by $e$, $u$ or 1. Thus, the unity satisfies

$e\cdot a\;=\;a\cdot e\;=\;a\quad\forall a\in R.$ |

If $R$ consists of the mere 0, then $0$ is its unity, since in every ring, $0\cdot a=a\cdot 0=0$. Conversely, if 0 is the unity in some ring $R$, then $R=\{0\}$ (because $a=0\cdot a=0\,\,\forall a\in R$).

Note. When considering a ring $R$ it is often mentioned that “…having $1\neq 0$” or that “…with non-zero unity”, sometimes only “…with unity” or “…with identity element”; all these exclude the case $R=\{0\}$.

###### Theorem.

An element $u$ of a ring $R$ is the unity iff $u$ is an idempotent and regular element.

Proof. Let $u$ be an idempotent and regular element. For any element $x$ of $R$ we have

$ux\;=\;u^{2}x\;=\;u(ux),$ |

and because $u$ is no left zero divisor, it may be cancelled from the equation; thus we get $x=ux$. Similarly, $x=xu$. So $u$ is the unity of the ring. The other half of the theorem is apparent.

## Mathematics Subject Classification

20-00*no label found*16-00

*no label found*13-00

*no label found*

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