Principal ideal

An ideal generated by a single element.

Principal ideal

A principal ideal in a commutative ring $R$ is an ideal of the form

(a)=\{ra:r\in R\}

for some $a\in R$ , i.e. an instance of an ideal generated by one element.

Principal ideals control divisibility and factorization in commutative algebra; rings in which every ideal is principal are PIDs . In noncommutative rings one often distinguishes left-principal and right-principal ideals.

Examples:

In $\mathbb Z$ , $(6)=6\mathbb Z$ is principal.
In $k[x,y]$ , $(x)$ is a principal ideal.
In $k[x,y]$ , the ideal $(x,y)$ is not principal.