Hauptidealringe

Hauptidealringe, or principal ideal rings, are rings in which every ideal is generated by a single element. In standard usage this term refers to commutative rings with identity, though some texts treat noncommutative cases as well. Formally, for every ideal I of a Hauptidealring R there exists a in R with I = (a).

Relation to principal ideal domains: A principal ideal domain (PID) is a Hauptidealring that is also an integral domain. Thus every PID is a principal ideal ring, but the converse is not true in general. For example, Z/nZ is a principal ideal ring for any n, but is not a domain when n is composite. Conversely, Z and the polynomial ring F[x] over a field F are both PIDs and hence principal ideal rings.

Examples and consequences: The ring of integers Z is a principal ideal domain and therefore a Hauptidealring. The ring F[x] of polynomials with coefficients in a field F is also a PID. Quotients of Hauptidealringe by any ideal are again Hauptidealringe, since ideals in the quotient correspond to principal ideals containing the quotient ideal and are generated by a single coset.

Properties: Principal ideal rings are Noetherian, because every ideal is finitely generated. They generalize domains: in the commutative case, a ring is a PID exactly when it is a Hauptidealring that is an integral domain. Related notions include Bezout domains, where every finitely generated ideal is principal in a domain.