unityUnity2004-11-02 12:30:572008-03-16 16:14:43Definitionleft and right unity of ringnon-zero unity% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}The {\em 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 \,$a = 0\cdot a = 0\,\,\forall a\in R$).
\textbf{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 \PMlinkescapetext{identity element}''; all these exclude the case \,$R = \{0\}$.
\begin{thmplain}
\, An element $u$ of a ring $R$ is the unity iff $u$ is an idempotent and regular element.
\end{thmplain}
{\em Proof.} \,Let $u$ be an idempotent and regular element. \,For any element $x$ of $R$ we have
$$ux = u^2x = u(ux),$$
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 \PMlinkescapetext{theorem} is apparent.