Modules

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As the notion of modules has been rediscovered in many areas,
modules have also been called autonomous sets, homogeneous sets, intervals,
and partitive sets.
Perhaps the earliest reference to them, and the first description of modular quotients
and the graph decomposition they give rise to appeared in (Gallai 1967).

A module of a graph is a generalization of a connected component.

Contrary to the connected components,
the modules of a graph are the same as the modules of its complement,
and modules can be "nested": one module can be a proper subset of another.
Note that the set V of vertices of a graph is a module,
as are its one-element subsets and the empty set; these are called the trivial modules.

A graph may or may not have other modules. A graph is called prime if all of its modules are trivial.

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In mathematics, a module is one of the fundamental algebraic structures used in
abstract algebra.

A module over a ring is a generalization of the notion of vector space over a field,
wherein the corresponding scalars are the elements of an arbitrary given ring
(with identity) and a multiplication (on the left and/or on the right) is defined
between elements of the ring and elements of the module.

A module, like a vector space, is an additive abelian group;
a product is defined between elements of the ring and elements of the module
that is distributive over the addition operation of each parameter and
is compatible with the ring multiplication.

Modules are very closely related to the representation theory of groups.
They are also one of the central notions of commutative algebra
and homological algebra, and are used widely in algebraic geometry
and algebraic topology.