Divisor (of an integer or of a polynomial)

2020 Mathematics Subject Classification: Primary: 13A05 [MSN][ZBL]

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).

More generally, in an arbitrary 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