View source for Divisor (of an integer or of a polynomial)

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''For other meanings of the term 'Divisor' see the page'' [[Divisor (disambiguation)]]

A divisor of an integer $a$ is an integer $b$ which divides $a$ without remainder. In other words, a divisor of the integer $a$ is an integer $b$ such that, for a certain integer $c$, the equality $a=bc$ holds.  A ''proper divisor'' or an [[aliquot divisor]] of $a$ is a natural number divisor of $a$ other than $a$ itself.

A divisor of a polynomial $A(x)$ is a polynomial $B(x)$ that divides $A(x)$ without remainder (cf. [[Division|Division]]). 

More generally, in an arbitrary [[Ring|ring]] $R$, a divisor of an element $a \in R$ is an element $b\in R$ such that $a=bc$ for a certain $c\in R$.

If $b\in R$ is a divisor of $a\in R$, one writes $b | a$.

If $a$ divides $b$ and $b$ divides $a$, then $a$ and $b$ are ''associates''.  If an element $a$ has the property that whenever $a = bc$, one of $b,c$ is an associate of $a$, then $a$ is ''irreducible''.  For polynomials, see [[Irreducible polynomial]]; for integers, the traditional terminology is [[prime number]].  

====References====
* David Sharpe, ''Rings and Factorization'' Cambridge University Press (1987) {{ISBN|0-521-33718-6}} {{ZBL|0674.13008}}