This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
This property of ideals in commutative unital rings depends only on the ideal, viewed abstractly as a module over the commutative unital ring. The corresponding module property that the ideal must satisfy is: cyclic module
View other such properties
Definition with symbols
An ideal in a ring is termed a principal ideal if there exists an in such that .
Relation with other properties
The whole ring, as well as the zero ideal, are principal ideals.
This property of ideals in commutative unital rings is not closed under taking arbitrary intersections; in other words, an arbitrary intersection of ideals with this property need not have this property
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.
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.