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If $a$ and $b$ are integers, $a$ divides $b$ if $b=ca$ for some integer $c$. This is denoted $a\mid b$. It is usually studied in introductory courses in number theory, so add (elementary-number-theory) tag, if appropriate.

A common notation used for the phrase "$a$ divides $b$" is $a|b$. It is also common to negate the notation by adding a slash like this: "$c$ does not divide $d$" written as $c\nmid d$. Note that the order is important: for example, $2|4$ but "$4\nmid 2$".

This notion can be generalized to any ring. The definition is the same: For two elements $a$ and $b$ of a commutative ring $R$, $a$ divides $b$ if $ac=b$ for some $c$ in $R$.

Divisibility in commutative rings corresponds exactly to containment the poset of principal ideals. That is, $a$ divides $b$ if and only if $aR\subseteq bR$. For commutative rings like principal ideal rings, this means that divisibility mirrors exactly the poset of all ideals of the ring.

The topics appropriate for this tag include, for example:

  • Questions about the relation $\mid$.
  • Questions about g.c.d and l.c.m.
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