Principal ideal

Symbol-free definition
An ideal in a commutative unital ring is termed a principal ideal if it is the ideal generated by a single element of the ring.

Definition with symbols
An ideal $$I$$ in a ring $$R$$ is termed a principal ideal if there exists an $$x$$ in $$R$$ such that $$I = Rx$$.

Weaker properties

 * Finitely generated ideal

Metaproperties
The whole ring, as well as the zero ideal, are principal ideals.

An intersection of principal ideals need not be a principal ideal. However, for a unique factorization domain, it is true that an arbitrary intersection of principal ideals is principal. Thus, given any ideal, there exists a smallest principal ideal containing it.

For a gcd domain, it is true that given any finitely generated ideal, there exists a smallest principal ideal containing it.

The property of being a principal ideal is not closed under taking finite, or arbitrary sums, of ideals. A finite sum of principal ideals is a finitely generated ideal, and every finitely generated ideal is principal iff the ring is a Bezout ring. Any ideal can be expressed as an arbitrary sum of principal ideals, and so an arbitrary sum of principal ideals is a principal ideal iff the ring is a principal ideal ring.