Subgenerators

Subgenerators is a term found in category theory and algebra, used to describe objects that help generate other objects through subobjects, or, in some contexts, as a dual notion to generators in module categories. The exact definition varies by author and context, so the term is not universally standardized.

In the module category Mod-R over a ring R, a common usage is: a right R-module G is a subgenerator if every right R-module M can be embedded into a product of copies of G; that is, there exists an index set I with an injective map M → G^I. In such texts, this property is closely related to the dual notion of a cogenerator, which typically requires every module to embed into a product of copies of a fixed object. Some authors explicitly call such objects cogenerators, and use subgenerator to emphasize the focus on subobjects rather than quotients.

Alternatively, some writers define a subgenerator more generally as an object that generates all objects of a particular class via subobjects (as opposed to generation via quotients, which is the domain of generators). In this view, a subgenerator provides a way to realize other objects inside powers or products of G, ensuring that G captures enough structure to represent subobjects of interest.

Examples and relationships:

- The regular module R_R is a generator for Mod-R because every module is a quotient of a free module (a direct sum of copies of R). Whether R_R is a subgenerator depends on the ring; for some rings the embedding property into a product of copies of R holds, for others it does not.

- In well-behaved categories (such as certain abelian categories), a subgenerator notion often aligns with the idea of a cogenerator.

See also: generator, cogenerator, abelian category, module category.