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.

Definition

Let ''R'' be a ring, and let ''a'' and ''b'' be elements of ''R''. If there exists an element ''x'' in ''R'' with , 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 , 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 $a = b u$ 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 $a$.

Zero as a divisor, and zero divisors

* If one interprets the definition of divisor literally, every ''a'' is a divisor of 0, since one can take . 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 .
* 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 , but such a definition is both more complicated and lacks some of the above properties.

See also

* Divisor – divisibility in integers
* – divisibility in polynomials
* Quasigroup – an otherwise generic magma with divisibility
* Zero divisor
* GCD domain

Notes

Citations

References

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Ring theory