Divisor (ring theory): Difference between revisions
imported>Barry R. Smith (added basic definition and properties) |
mNo edit summary |
||
| (6 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | {{subpages}} | ||
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. Divisibility is a useful concept for the analysis of the structure of commutative rings | 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. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings. | ||
==Definition== | ==Definition== | ||
| Line 7: | Line 7: | ||
A nonzero element ''b'' of a commutative ring ''R'' is said to divide an element ''a'' in ''R'' (notation: <math> a \mid b </math>) if there exists an element ''x'' in ''R'' with <math> a = b x </math>. We also say that ''b'' is a divisor of ''a'', or that ''a'' is a [[multiple]] of [[b]]. | A nonzero element ''b'' of a commutative ring ''R'' is said to divide an element ''a'' in ''R'' (notation: <math> a \mid b </math>) if there exists an element ''x'' in ''R'' with <math> a = b x </math>. We also say that ''b'' is a divisor of ''a'', or that ''a'' is a [[multiple]] of [[b]]. | ||
Notes: This definition makes sense when ''R'' is any commutative [[ | Notes: This definition makes sense when ''R'' is any commutative [[semigroup]], but virtually the only time divisors are discussed is when this semi-group is the multiplicative [[monoid]] of a commutative ring with identity. Also, divisors are also occasionally useful in non-commutative contexts, where one must then discuss [[right divisor|left]] and [[right divisor]]s. | ||
Elements ''a'' and ''b'' of a commutative ring are said to be '''associates''' if both <math> a \mid b </math> and <math> b \mid a </math>. The associate relationship is an [[equivalence relation]] on 'R', and hence divides 'R' into [[disjoint]] [[equivalence | Elements ''a'' and ''b'' of a commutative ring are said to be '''associates''' if both <math> a \mid b </math> and <math> b \mid a </math>. The associate relationship is an [[equivalence relation]] on ''R'', and hence divides ''R'' into [[disjoint sets|disjoint]] [[equivalence class]]es each of which consists of all elements of ''R'' that are associates of any particular member of the class. | ||
==Properties== | ==Properties== | ||
If ''R'' has an identity, then most statements about divisibility can be translated into statements about principal ideals. For instance, | |||
* <math> b \mid a </math> if and only if <math> (a) \subset (b) </math>. | * <math> b \mid a </math> if and only if <math> (a) \subset (b) </math>. | ||
* 'a' and 'b' are associates if and only if <math> (a) = (b) </math> | * ''a'' and ''b'' are associates if and only if <math> (a) = (b) </math> | ||
* ''u'' is a unit if and only if ''u'' is a divisor of every element of ''R'' | |||
* ''u'' is a unit if and only if <math> (u) = R </math>. | |||
* If <math> a = b u </math> where ''u'' is a unit, then ''a'' and ''b'' are associates. If ''R'' is an [[integral domain]], then the converse is true.[[Category:Suggestion Bot Tag]] | |||
* 'u' is a unit if and only if <math> (u) = R </math>.If <math> a = b u </math> where ''u'' is a unit, then ''a'' and ''b'' are associates. If 'R' is an [[integral domain]], then the converse is true. | |||
Latest revision as of 06:00, 8 August 2024
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. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings.
Definition
A nonzero element b of a commutative ring R is said to divide an element a in R (notation: ) if there exists an element x in R with . We also say that b is a divisor of a, or that a is a multiple of b.
Notes: This definition makes sense when R is any commutative semigroup, but virtually the only time divisors are discussed is when this semi-group is the multiplicative monoid of a commutative ring with identity. Also, divisors are also occasionally useful in non-commutative contexts, where one must then discuss left and right divisors.
Elements a and b of a commutative ring are said to be associates if both and . The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes each of which consists of all elements of R that are associates of any particular member of the class.
Properties
If R has an identity, then most statements about divisibility can be translated into statements about principal ideals. For instance,
- if and only if .
- a and b are associates if and only if
- u is a unit if and only if u is a divisor of every element of R
- u is a unit if and only if .
- If where u is a unit, then a and b are associates. If R is an integral domain, then the converse is true.
