Associate elements in non-integral domains. [duplicate]
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If $R$ is a commutative ring with identity, and $a, bin R$ are divisible by each other, is it true that they must be associates?
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Elements $a$ and $b$ of an integral domain are associates if $amathrel{vdots}b$ and $bmathrel{vdots}a$ I have proved this fact. Then I tried to find out $a$ and $b$ which divide each other and does not associate. This may happen only in non-integral domains of course. But I couldn't find an example. Can someone suggest one? Definition of associates. $a$ and $b$ are associates if $a=bepsilon$ where $epsilon$ is invertible element of the ring. $amathrel{vdots}b$ means $a$ is divide...