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| 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 |
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 |
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$$e\cdot a = a\cdot e = a\quad\forall a\in R.$$
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$$ea = ae = a\quad\forall a\in R.$$
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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 \,$a = 0\cdot a = 0\,\,\forall a\in R$).
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If $R$ consists of the mere 0, then $0$ is its unity, since in every ring, \,$0a = a0 = 0$. \,Conversely, if 0 is the unity in some ring $R$, then \,$R = \{0\}$\, (because \,$a = 0a = 0\,\,\forall a\in R$).
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| 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 \,$R = \{0\}$. |
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 \,$R = \{0\}$. |
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