Monomodules

Monomodules are a concept in abstract algebra, specifically within the study of modules over a ring. A module is a generalization of a vector space where the scalars are from a ring rather than a field. A monomodule is a module that can be generated by a single element. This means that for a module M over a ring R, M is a monomodule if there exists an element g in M such that every element m in M can be expressed as r * g for some scalar r from the ring R. The set of all such elements is precisely M itself.

In simpler terms, a monomodule is a module where all its elements are multiples of a single element within the module. This is analogous to a cyclic group in group theory, where all elements are powers of a single generator. Examples of monomodules include cyclic subgroups of additive groups and principal ideals of a ring. The structure of monomodules is relatively simple and depends heavily on the properties of the ring R. If R is a field, then any non-zero R-module is a monomodule, isomorphic to R itself as a vector space. For general rings, the structure can be more complex, and the study of monomodules is often related to the study of principal ideals and the structure of the ring itself.