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> Suppose we have a ring $R$ and $(a),(b)$ are both ideals of $R$. Is it always true that $(a)=(b)$ if and only if there exists a unit $c$ such that $a=bc$ (i.e., $a$ and $b$ are associate)? 
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I have already verified that the forwards direction is true. But I have no idea on the backward direction. If it is true, can someone provide a proof to me?

Backward direction: Suppose that there exists a unit $c \in R$ such that $a=bc$. This implies that $(a)\subset (b)$. By using the same thing, (i.e., $b$ and $a$ are associate), there exists a unit $d \in R$ such that $b=ad$. This implies that $(b) \subset (a)$. is this correct?