24 lines
629 B
TeX
24 lines
629 B
TeX
\section{Properties of Rings}
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\begin{theorem}[Rules of Multiplication]
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Let $a,b$, and $c$ belong to a ring $R$. Then
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\begin{enumerate}
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\item $a0 = 0a = 0$.
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\item $a(-b) = (-a)b = -(ab)$.
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\item $(-a)(-b) = ab$.
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\item $a(b-c) = ab - ac$ and $(b-c)a = ba - ca$.
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\end{enumerate}
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Furthermore, if $R$ has a unity element $1$, then
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\begin{enumerate}
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\setcounter{enumi}{4}
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\item $(-1)a = -a$.
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\item $(-1)(-1) = 1$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[Uniqueness of the Unity and Inverses]
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If a ring has a unity, it is unique. If a ring element has a multiplicative inverse, it is unique.
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\end{theorem}
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