Divisibility (ring theory)

In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. 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.

Examples

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

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

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

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

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

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

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

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

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

Definition

Let R be a ring, 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. 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 $a\mid b$. Elements a and b of an integral domain are associates if both $a\mid b$ and $b\mid a$. 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.

Properties

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

• One has $a\mid b$ if and only if $(b)\subseteq (a)$.
• 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 $a=bu$ 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, $(a)$ denotes the principal ideal of $R$ generated by the element $a$.

Zero as a divisor, and zero divisors

• 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.

See also

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

References

• 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.