monogen

Monogen is a term that appears in scholarly writing as an abbreviation or variant of monogenic, depending on context. It is not a widely recognized standalone term with a single canonical definition, and in most mathematical literature the full form monogenic is preferred. When used, monogen often signals that a structure is generated by a single element.

In mathematics, monogen refers to objects that can be generated by one element over a base structure. For example, a ring R is monogenic over a subring S if R = S[a] for some element a in R. A field extension K over a base field F is monogenic if K = F(α) for some α. In algebraic number theory, whether a given ring of integers is monogenic (generated by one element as a Z-algebra) is a classical problem, with many rings proven to be non-monogenic.

In group theory, the concept aligns with the idea of a cyclic group, which is generated by a single element and is thus monogenic in the sense of being produced by one generator.

In biology or parasitology, related terms exist but the standalone term monogen is not standard. Monogenea is a class of parasitic flatworms, and monogenic, in genetic discussions, is more commonly used to describe a trait controlled by a single gene. Readers should consult the specific discipline’s terminology to determine the intended meaning.

Etymology: the term derives from Greek monos, meaning one, and gen, meaning origin or generator. The usage of monogen tends to be context-dependent and largely superseded by monogenic in formal writing.

See also: monogenic, simple extension, cyclic group, monogenea.