féldefinite

Féldefinite is a term used in abstract algebra, specifically in the study of rings and modules. It describes a property of certain algebraic structures. A ring R is said to be féldefinite if every finitely generated ideal of R is principal. This means that for any ideal I that can be generated by a finite set of elements, there exists a single element g in R such that I is equal to the set of all multiples of g within R.

This property is a generalization of the concept of a principal ideal domain (PID), where every ideal, not just those that are finitely generated, must be principal. Therefore, every PID is féldefinite. However, the converse is not always true; a ring can be féldefinite without being a PID. An example of such a ring is a Dedekind domain that is not a PID.

The féldefinite property is significant because it ensures a certain level of structure and predictability within the ring. It simplifies many algebraic constructions and proofs. For instance, in a féldefinite ring, concepts like greatest common divisors and least common multiples behave in a more straightforward manner, similar to how they do in the familiar setting of integers. The study of féldefinite rings is an active area of research in commutative algebra.