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An additive subgroup closed under the scalar action of a module.
Let be a left -module. A submodule of is a subset such that:
Equivalently, is an additive subgroup of that is stable under scalar multiplication. Practical closure tests are summarized in the submodule criterion.
Let be a ring (often assumed a unital ring). A left -module is an abelian group together with a scalar multiplication map , , such that for all and :
A right -module is defined similarly with a map , , satisfying the analogous axioms.
The axioms are collected in module axioms. When is a field, left -modules are the same objects as vector spaces. Ideals of a ring give basic examples of modules, linking module theory to ideal theory.
A subset of a set is a set such that every element of is an element of :
Submodule criterion: Let be an -module and let be a subset. Then is a submodule of if and only if:
This is the standard test for submodules obtained by unpacking the module axioms for modules over a ring.
Let be a ring and an additive subgroup. Then is a left ideal if for every , a right ideal if for every , and a two-sided ideal if both conditions hold.
A (left) -module is simple if its only submodules are and .
Simple modules are the analogs of "atoms" in module theory: they appear as factors in composition series, and semisimplicity is built from direct sums of simples. Over a commutative ring, simple modules are precisely where is a maximal ideal, hence are vector spaces over the field (compare quotients).