Submodules are the internal pieces of a [module](/page/Module) that still remember the full scalar action. A module is an additive abelian group equipped with multiplication by elements of a [ring](/page/Ring); a submodule is the kind of subset on which both operations remain valid. This is the module-theoretic analogue of a [subgroup](/page/Subgroup) inside a group, a [subspace](/page/Vector%20Space) inside a vector space, and, for the regular left module of a ring over itself, a left [ideal](/page/Ideal) inside a ring. Without submodules, quotient modules, exact sequences, kernels, images, generators, and most of homological algebra would have no natural internal language.

The definition is shaped by a closure problem. If $M$ is an $R$-module and $N \subset M$, then $N$ can only be treated as a module in its own right if sums, additive inverses, and scalar multiples of elements of $N$ stay inside $N$. Closure under addition alone is not enough, because the scalar action may leave the subset. Closure under scalar multiplication alone is not enough, because the additive group structure may fail.

## Definition

The first definition isolates exactly the closure conditions needed for a subset to inherit the module structure from its ambient module. It points back to the parent notion of module: a submodule is not an unrelated object inside $M$, but a subset on which the same module operations restrict.

[definition: Submodule] Let $R$ be a ring and let $M$ be a left $R$-module. A subset $N \subset M$ is an $R$-submodule of $M$, written $N \le M$, if: 1. $N$ is a subgroup of the additive group $(M,+)$; 2. for every $r \in R$ and every $n \in N$, one has $rn \in N$. [/definition]

The notation $N \le M$ deliberately parallels subgroup notation. It records more than inclusion: it says that $N$ is closed under the relevant algebraic structure. For modules over a fixed ring $R$, the ambient module usually determines the scalar action, so authors often say simply that $N$ is a submodule of $M$.

The formulae on this page use the standard algebra convention that rings have a multiplicative identity and modules are unital. This matters most for generated submodules: under this convention, $1_Rm=m$, so the set of scalar multiples of one element already contains the generator itself.

[remark: Ring and Module Convention] Throughout this page, rings are assumed to have a multiplicative identity $1_R$, and left or right $R$-modules are assumed to be unital: $1_Rm=m$ for every module element $m$. [/remark]

When the ring is noncommutative, the side of the scalar action becomes part of the structure. A subset stable under left multiplication need not be stable under right multiplication, so the right-handed version deserves its own explicit definition.

[definition: Right Submodule] Let $R$ be a ring and let $M$ be a right $R$-module. A subset $N \subset M$ is a right $R$-submodule of $M$ if: 1. $N$ is a subgroup of the additive group $(M,+)$; 2. for every $n \in N$ and every $r \in R$, one has $nr \in N$. [/definition]

This page focuses on left modules unless stated otherwise. The right-module version is obtained by moving the scalar to the right throughout the statements.

Many arguments distinguish the whole module from a genuine smaller internal piece. That distinction is needed for maximal submodules, simple quotients, and induction on subobjects, so the strict version is named separately.

[definition: Proper Submodule] Let $R$ be a ring, let $M$ be an $R$-module, and let $N \le M$. The submodule $N$ is a proper submodule of $M$ if $N \subsetneq M$. [/definition]

Proper submodules are where maximality and simplicity enter the subject. A module with no nonzero proper submodules behaves like an indivisible object, while a module with many proper submodules has internal structure that can be studied through quotients and filtrations.

Once submodules have been named, the next natural question is how all of them sit inside a fixed module. Organising them by containment makes intersections, sums, generated submodules, and quotient constructions part of one ordered framework.

[definition: Submodule Lattice] Let $R$ be a ring and let $M$ be an $R$-module. The submodule lattice of $M$ is the set \begin{align*} \operatorname{Sub}_R(M) := \{N \subset M : N \le M\}, \end{align*} ordered by inclusion. [/definition]

The word "lattice" is justified by the intersection and sum operations developed below. This ordered viewpoint is especially important in representation theory, where invariant subspaces and composition series are organised by containment.

## Equivalent Characterisations

Checking the subgroup condition directly can be more work than necessary. Since a module already has an abelian group structure, there is a compact test using subtraction and scalar multiplication. This test is often the fastest way to verify that a candidate subset is a submodule.

[quotetheorem:8497]