Editing Least common multiple (section)

== In commutative rings ==
The least common multiple can be defined generally over [[commutative ring]]s as follows:

Let {{mvar|a}} and {{mvar|b}} be elements of a commutative ring {{mvar|R}}. A ''common multiple'' of {{mvar|a}} and {{mvar|b}} is an element {{mvar|m}} of {{mvar|R}} such that both {{mvar|a}} and {{mvar|b}} divide {{mvar|m}} (that is, there exist elements {{mvar|x}} and {{mvar|y}} of {{mvar|R}} such that {{math|''ax'' {{=}} ''m''}} and {{math|''by'' {{=}} ''m''}}). A ''least common multiple'' of {{mvar|a}} and {{mvar|b}} is a common multiple that is minimal, in the sense that for any other common multiple {{mvar|n}} of {{mvar|a}} and {{mvar|b}}, {{mvar|m}} divides&nbsp;{{mvar|n}}.

In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are [[Unit (ring theory)|associates]].{{sfn|Burton|1970|p=94}} In a [[unique factorization domain]], any two elements have a least common multiple.{{sfn|Grillet|2007|p=142}} In a [[principal ideal domain]], the least common multiple of {{mvar|a}} and {{mvar|b}} can be characterised as a generator of the intersection of the ideals generated by {{mvar|a}} and {{mvar|b}}{{sfn|Burton|1970|p=94}} (the intersection of a collection of ideals is always an ideal).