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# If a = bc with $a \neq 0$ and b and c nonunits, show that a is not an associate of b.

Solution

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$\textbf{Given}$: Let $R$ is an integral domain such that $\>a=bc\>$ for some elements $a \neq 0$ and $b,c$ are non-units in $R$.

$\textbf{To Prove}$: $a$ is not an associate of $b$.

$\textbf{Proof}$: If possible, let us assume that $a$ is an associate of $b$ in $R$. Then there exists an unit $u$ in $R$ such that

$a=bu.$

Now by the given condition note that

\begin{align*} a=bc \>\> \text{ and }\> a=bu \implies \> & bc=bu \\ \implies \> & b(c-u)=0 \\ \implies \> & c=u,\>\> \text{ since R is an integral domain and a \neq 0 in }R.\end{align*}

This contradicts the fact that $c$ is a non unit in $R$. So our assumption that, $a$ is an associate of $b$ in $R$ is wrong. Hence, $a$ is not an associate of $b$ in $R$. This completes the solution.

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