Divisor (ring theory): Difference between revisions
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In [[mathematics]], the notion of a '''divisor''' originally arose within the context of arithmetic of whole numbers. Please see the page about [[divisor]]s for this simplest example. With the development of abstract [[ | In [[mathematics]], the notion of a '''divisor''' originally arose within the context of arithmetic of whole numbers. Please see the page about [[divisor]]s for this simplest example. With the development of abstract [[Ring (mathematics)|rings]], of which the integers are the archetype, the original notion of divisor found a natural extension. The definition makes sense for any commutative [[semi-group]] (and in fact, notions of left and right divisors are used in non-commutative contexts), but divisors are most useful within the confines of commutative rings. The reason is that the notion of divisibility within a commutative ring is related to the notion of set containment of [[ideal]]s, and many useful properties about divisibility can be translated into statements about ideals. | ||
Revision as of 15:54, 29 March 2008
In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. Please see the page about divisors for this simplest example. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. The definition makes sense for any commutative semi-group (and in fact, notions of left and right divisors are used in non-commutative contexts), but divisors are most useful within the confines of commutative rings. The reason is that the notion of divisibility within a commutative ring is related to the notion of set containment of ideals, and many useful properties about divisibility can be translated into statements about ideals.