13-00-UnitiesOfRingAndSubring.tex

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\pmtitle{unities of ring and subring}
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\begin{document}
Let $R$ be a ring and $S$ a proper subring of it. \,Then there exists five cases in all concerning the possible unities of $R$ and $S$.
\begin{enumerate}
 \item $R$ and $S$ have a common unity.
 \item $R$ has a unity but $S$ does not.
 \item $R$ and $S$ both have their own non-zero unities but these are distinct.
 \item $R$ has no unity but $S$ has a non-zero unity.
 \item Neither $R$ nor $S$ have unity.
\end{enumerate}
\textbf{Note:} \, In the cases 3 and 4, the unity of the subring $S$ must be a zero divisor of $R$.

\textbf{Examples}
\begin{enumerate}
 \item The ring $\mathbb{Q}$ and its subring $\mathbb{Z}$ have the common 
unity 1.
 \item The subring $S$ of even integers of the ring $\mathbb{Z}$ has no unity.
 \item Let $S$ be the subring of all pairs $(a,\,0)$ of the ring 
\,$R = \mathbb{Z}\times\mathbb{Z}$\, for which the operations ``$+$'' and ``$\cdot$'' are defined componentwise (i.e. \,$(a,\,b)+(c,\,d) = (a+c,\,b+d)$\, etc.). \,Then $S$ and $R$ have the unities $(1,\,0)$ and $(1,\,1)$, respectively.
 \item Let $S$ be the subring of all pairs $(a,\,0)$ of the ring \,$R = \{(a,\,2b)|\,\,\,a\in\mathbb{Z}\,\land \,b\in\mathbb{Z}\}$ (operations componentwise). \,Now $S$ has the unity $(1,\,0)$ but $R$ has no unity.
 \item Neither the ring $\{(2a,\,2b)|\,\,\,a,\,b\in\mathbb{Z}\}$ (operations componentwise) nor its subring consisting of the pairs $(2a,\,0)$ have unity.
\end{enumerate}
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