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2016-07-20-rings_02.tex
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2016-07-20-rings_02.tex
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\DiaryEntry{Rings - Examples}{2016-07-20}{Algebra}
\subsection{Example $\mZ / 5\mZ$}
As an example consider the ring \(\mathbb{Z}/5\mathbb{Z}\) with the following table for addition (mod-5)
\[
\begin{array}{c|ccccc}
+ & 0 & 1 & 2 & 3 & 4 \\
\hline
0 & 0 & 1 & 2 & 3 & 4 \\
1 & 1 & 2 & 3 & 4 & 0 \\
2 & 2 & 3 & 4 & 0 & 1 \\
3 & 3 & 4 & 0 & 1 & 2 \\
4 & 4 & 0 & 1 & 2 & 3 \\
\end{array}
\]
Every row contains exactely one zero; in other words, every element has exactely one additive inverse and the set with addition forms an abelian group.
The multiplication (mod-5) table looks like this
\[
\begin{array}{c|ccccc}
\times & 0 & 1 & 2 & 3 & 4 \\
\hline
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 2 & 3 & 4 \\
2 & 0 & 2 & 4 & 1 & 3 \\
3 & 0 & 3 & 1 & 4 & 2 \\
4 & 0 & 4 & 3 & 2 & 1 \\
\end{array}
\]
Every row (except zero) contains exactely one element 1; this implies that every element apart from zero has exactely one multiplicative inverse.
The set is actually not only a ring but a field as every element has a multiplicative inverse.
\subsection{Example $\mZ/6\mZ$}
As an example of a ring consider the field \(\mathbb{Z}/6\mathbb{Z}\) with the following table for addition
\[
\begin{array}{c|cccccc}
+ & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
0 & 0 & 1 & 2 & 3 & 4 & 5 \\
1 & 1 & 2 & 3 & 4 & 5 & 0 \\
2 & 2 & 3 & 4 & 5 & 0 & 1 \\
3 & 3 & 4 & 5 & 0 & 1 & 2 \\
4 & 4 & 5 & 0 & 1 & 2 & 3 \\
5 & 5 & 0 & 1 & 2 & 3 & 4 \\
\end{array}
\]
Every element has an additive inverse;i.e. the set forms a group with respect to addition. The multiplication table has the following form
\[
\begin{array}{c|cccccc}
\times & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 2 & 3 & 4 & 5 \\
2 & 0 & 2 & 4 & 0 & 2 & 4 \\
3 & 0 & 3 & 0 & 3 & 0 & 3 \\
4 & 0 & 4 & 2 & 0 & 4 & 2 \\
5 & 0 & 5 & 4 & 3 & 2 & 1 \\
\end{array}
\]
The rows with element 2, 3, and 4 have more than one zero; i.e.~the element 2 has two multiplicative inverses, namely 3 and 5, because \(2\times3=2\times5=0\). It is exactely the rows with value \(k\) for
which \(\gcd(k,6) \neq 1\) for which no unique multiplicative inverse exists.
However, $\mZ/6\mZ$ is a ring; the set with multiplication need not be a group (and therefore have aninverse).