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Let $R$ be an integral domain with identity.

A unit of $R$ is an element $u \in R$ which divides 1.

Does this mean every element in $\mathbb R$ (real numbers) is a unit since every element divides 1?

e.g.: $3$ is a unit since $\frac{1}{3} \in \mathbb R$

Also, two elements $a,b$ are called associates if there is a unit $u$ such that $a = bu$. Doesn't this also mean every element in $\mathbb R$ is associated with every other element?

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Almost yes. In any field, all elements but 0 are units and therefore associated with each other. – martini Nov 7 '12 at 13:43
up vote 4 down vote accepted

Being associate is an equivalence relation. In any field, there are exactly two equivalence classes for this relation: the class of 0 and the class of 1. The class of 0 contains just 0; the class of 1 contains everything else.

So, in a field, every two nonzero elements are associates.

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