Divisibility (ring theory)

In the ring of integers ${\displaystyle R=\mathbb {Z} }$ consider elements ${\displaystyle a,b}$.

• ${\displaystyle a=4}$ is a divisor of ${\displaystyle b=12}$, because ${\displaystyle 12\div 4=3}$ with remainder ${\displaystyle 0}$ or, phrased differently and more appropriate for generalization, there is ${\displaystyle x=3\in R}$ such that ${\displaystyle ax=b}$.
• ${\displaystyle a=4}$ is not a divisor of ${\displaystyle b=13}$, because ${\displaystyle 13\div 4=3}$ with remainder ${\displaystyle 1}$ resp. there is no ${\displaystyle x\in R}$ such that ${\displaystyle ax=b}$.

In the field of reals ${\displaystyle S=\mathbb {R} }$, an extension of the integers, we have

• ${\displaystyle a=4}$ is still a divisor of ${\displaystyle b=12}$, because ${\displaystyle 12\div 4=3}$ resp. there is ${\displaystyle x=3\in S}$ such that ${\displaystyle ax=b}$.
• ${\displaystyle a=4}$ becomes a divisor of ${\displaystyle b=13}$, because ${\displaystyle 13\div 4=3.25}$ is now a real number in ${\displaystyle S}$ resp. there is ${\displaystyle x=3.25\in S}$ such that ${\displaystyle ax=b}$.

In the ring of integer polynomials ${\displaystyle R'=\mathbb {Z} [X]}$ we have that

• ${\displaystyle a=X+1}$ is a divisor of ${\displaystyle b=X^{2}-1}$, because by polynomial division ${\displaystyle (X^{2}-1)\div (X+1)=X-1}$ with remainder ${\displaystyle 0}$ resp. there is ${\displaystyle x=X-1\in R'}$ such that ${\displaystyle ax=b}$.
• ${\displaystyle a=X+3}$ is not a divisor of ${\displaystyle b=X^{2}-3}$, because ${\displaystyle (X^{2}-3)\div (X+3)=X-3}$ with remainder ${\displaystyle 6}$ resp. there is no ${\displaystyle x\in R'}$ such that ${\displaystyle ax=b}$. In fact one can show that ${\displaystyle b=X^{2}-3}$ has no nontrivial divisors, which in this case means that the quadratic polynomial ${\displaystyle b}$ has no linear divisors like ${\displaystyle a=X+3}$.

In the ring of real polynomials ${\displaystyle S'=\mathbb {R} [X]}$, an extension of the integer polynomials, we have

• ${\displaystyle a=X+1}$ is still a divisor of ${\displaystyle b=X^{2}-1}$, because ${\displaystyle (X^{2}-1)\div (X+1)=X-1}$ resp. there is ${\displaystyle x=X-1\in S'}$ such that ${\displaystyle ax=b}$.
• ${\displaystyle b=X^{2}-3}$ acquires nontrivial divisors, explicitly ${\displaystyle {\tilde {a}}=X+{\sqrt {3}}}$, because ${\displaystyle (X^{2}-3)\div (X+{\sqrt {3}})=X-{\sqrt {3}}}$ resp. there is ${\displaystyle {\tilde {x}}=X-{\sqrt {3}}\in S'}$ such that ${\displaystyle {\tilde {a}}{\tilde {x}}=b}$. Note that ${\displaystyle a=X+3}$ is still not a divisor of ${\displaystyle b=X^{2}-3}$, because of the same reason as for ${\displaystyle R'=\mathbb {Z} [X]}$ above.

These examples illustrate that the notion of divisibility does not only depend on the elements ${\displaystyle a}$ and ${\displaystyle b}$ themselves, but also on the context of the algebraic structure of a ring in which ${\displaystyle a,b}$ and the multiplication operation are considered. In general divisibility is preserved under extensions and more divisibility may be acquired.

Let R be a ring,[1] and let a and b be elements of R. If there exists an element x in R with ax = b, one says that a is a left divisor of b and that b is a right multiple of a.[2] Similarly, if there exists an element y in R with 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 ${\displaystyle a\mid b}$. Elements a and b of an integral domain are associates if both ${\displaystyle a\mid b}$ and ${\displaystyle b\mid a}$. 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 ${\displaystyle R}$ can be translated into statements about principal ideals. For instance,

• One has ${\displaystyle a\mid b}$ if and only if ${\displaystyle (b)\subseteq (a)}$.
• Elements a and b are associates if and only if ${\displaystyle (a)=(b)}$.
• 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 ${\displaystyle (u)=R}$.
• If ${\displaystyle a=bu}$ 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, ${\displaystyle (a)}$ denotes the principal ideal of ${\displaystyle R}$ generated by the element ${\displaystyle a}$.

• Some authors require a to be nonzero in the definition of divisor, but this causes some of the properties above to fail.
• If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take 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 ax = 0.[3]

• Divisors of Integers
• Divisibility of polynomials
• Zero divisor
• GCD domain

• Bourbaki, N. (1989) [1970], Algebra I, Chapters 1–3, Springer-Verlag, ISBN 9783540642435

This article incorporates material from the Citizendium article "Divisibility (ring theory)", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.