Divisibility (ring theory)#Definition

Let R be a ring,{{efn|In this article, rings are assumed to have a 1.}} and let a and b be elements of R. If there exists an element x in R with {{nowrap|1=ax = b}}, one says that a is a left divisor of b and that b is a right multiple of a.{{sfn|Bourbaki|1989|p=97|ps=none}} Similarly, if there exists an element y in R with {{nowrap|1=ya = b}}, one says that a is a right divisor of b and that b is a left multiple of a. One says that a is a two-sided divisor of b if it is both a left divisor and a right divisor of b; the x and y above are not required to be equal.

When R is commutative, the notions of left divisor, right divisor, and two-sided divisor coincide, so one says simply that a is a divisor of b, or that b is a multiple of a, and one writes $a\; \backslash mid\; b$. Elements a and b of an integral domain are associates if both $a\; \backslash mid\; b$ and $b\; \backslash mid\; a$. The associate relationship is an equivalence relation on R, so it divides R into disjoint equivalence classes.

Note: Although these definitions make sense in any magma, they are used primarily when this magma is the multiplicative monoid of a ring.

Statements about divisibility in a commutative ring $R$ can be translated into statements about principal ideals. For instance,

• One has $a\; \backslash mid\; b$ if and only if $(b)\; \backslash subseteq\; (a)$.
• Elements a and b are associates if and only if $(a)\; =\; (b)$.
• An element u is a unit if and only if u is a divisor of every element of R.
• An element u is a unit if and only if $(u)\; =\; R$.
• If $a\; =\; b\; u$ for some unit u, then a and b are associates. If R is an integral domain, then the converse is true.
• Let R be an integral domain. If the elements in R are totally ordered by divisibility, then R is called a valuation ring.

In the above, $(a)$ denotes the principal ideal of $R$ generated by the element $a$.

• If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take {{nowrap|1=x = 0}}. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that {{nowrap|1=ax = 0}}.{{sfn|Bourbaki|1989|p=98|ps=none}}
• Some texts apply the term 'zero divisor' to a nonzero element x where the multiplier a is additionally required to be nonzero where x solves the expression {{nowrap|1=ax = 0}}, but such a definition is both more complicated and lacks some of the above properties.

• Divisor – divisibility in integers
• {{slink|Polynomial#Divisibility}} – divisibility in polynomials
• Quasigroup – an otherwise generic magma with divisibility
• Zero divisor
• GCD domain

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• {{citation |last=Bourbaki |first=N. |author-link=Nicolas Bourbaki |year=1989 |orig-year=1970 |title=Algebra I, Chapters 1–3 |publisher=Springer-Verlag |isbn=9783540642435 |url=https://books.google.com/books?id=STS9aZ6F204C }}

{{Citizendium|Divisor (ring theory)}}

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Category:Ring theory