Associate elements

Definition

Let $R$ be a commutative ring and let $a, b \in R$. The elements $a$ and $b$ are associates, written $a \sim b$, if there is a unit $u \in R$ with $b = u a$.

Remarks

Being associates is an equivalence relation. In an integral domain, $a \sim b$ if and only if $a \mid b$ and $b \mid a$, if and only if $(a) = (b)$. In $\mathbb{Z}$, the associates of $n$ are $n$ and $-n$. In $\mathbb{Z}[i]$, the associates of $1 + i$ are $1 + i$, $-1 - i$, $-1 + i$ and $1 - i$.

Explanations

L1 human

Associates are elements that differ only by a unit factor. For divisibility they behave in the same way. In the integers, $6$ and $-6$ are associates.