Monogenic

Monogenic is an adjective used in several fields of mathematics and science to describe a structure that is generated or determined by a single element. The term comes from the Greek roots mono- (one) and genein (to be born or produced).

In algebra, a left module M over a ring R is monogenic if it can be generated by a single element; that is, M = Rm for some m in M. Such modules are also called cyclic. The concept extends to algebras: a monogenic algebra over a subring is generated by one element, often written as R[α]. This single-element origin underpins many basic constructions and classifications in module and ring theory.

In algebraic number theory, a number field K is called monogenic if its ring of integers O_K is generated by a single element as a Z-module, i.e., O_K = Z[θ] for some θ. Not all number fields have this property, and determining monogenicity can be subtle, involving index forms and discriminants. The topic connects to broader questions about how arithmetic invariants arise from simple generators.

In Clifford analysis, a branch of higher-dimensional function theory, a function is monogenic if it is in the kernel of the Dirac operator (D f = 0). Monogenic functions generalize holomorphic functions to multiple dimensions and obey analogues of Cauchy’s integral formulas and power series.

In genetics, monogenic describes traits or diseases caused by variation in a single gene. Monogenic disorders are inherited according to Mendelian patterns and contrast with polygenic or multifactorial traits. Examples include certain cystic fibrosis mutations and Huntington’s disease, though many conditions are influenced by multiple genes and environmental factors.