Example

(1) In any module $E$ the set consisting of 0 is a submodule (the zero submodule, often denoted by 0, by an abuse of notation). (2) Let $A$ be a ring. The submodules of $A_s$ (resp. $A_d$) are just the left ideals (resp. right ideals) of the ring $A$. (3) Let $E$ be an $A$-module, $x$ an element of $E$ and $a$ a left ideal of $A$. The set of elements $\alpha x$, where $\alpha$ runs through $a$, is a submodule of $E$, denoted by $ax$. (4) In a commutative group $G$ considered as a $\mathbf{Z}$-module (no. 1), every subgroup of $G$ is also a submodule.

(5) Let $I$ be an open interval of the real line $\mathbf{R}$; the set $C$ of real-valued functions defined and continuous on $I$ is a vector subspace of the vector space $\mathbf{R}^I$ of all real-valued functions defined on $I$. Similarly, the set $D$ of differentiable functions on $I$ is a vector subspace of $C_*$

Let $E$ be an $A$-module. Every equivalence relation compatible (I, § 1, no. 6) with the module structure on $E$ is of the form $x - y \in M$, where $M$ is a stable subgroup of E (I, § 4, no. 4), that is a submodule of E. It is immediately verified that the structure of a group with operators on the quotient group E/M (I, § 4, no. 4) is an A-module structure, under which the canonical mapping $E \to E/M$ is linear; with this structure, E/M is called the quotient module of E by the submodule M. A quotient module of a vector space E is called a quotient vector space (or simply quotient space) of E.

Example (6). Every left ideal $a$ in a ring A defines a quotient module $A_s/a$ of the left A-module $A_s$; by an abuse of notation, this quotient module is often denoted by $A/a$.

Let E, F be two A-modules. It follows from the general properties of groups with operators (I, § 4, no. 5) (or directly from the definitions) that if $u : E \to F$ is a linear mapping, the image under $u$ of any submodule of E is a submodule of F and the inverse image under $u$ of any submodule of F is a submodule of E. In particular, the kernel $N = \overline{u}^{-1}(0)$ is a submodule of E and the image $u(E)$ of E under $u$ is a submodule of F (I, § 4, no. 6, Proposition 7); by an abuse of language $u(E)$ is called the image of u. The quotient module $E/N$ is also called the coimage of $u$ and the quotient module $F/u(E)$ the cokernel of $u$. In the canonical decomposition of $u$ (I, § 4, no. 5)

$v$ is an isomorphism of the coimage of $u$ onto the image of $u$ (no. 2, Proposition 2). For $u$ to be injective, it is necessary and sufficient that its kernel be zero; for $u$ to be surjective, it is necessary and sufficient that its cokernel be zero.

The kernel, image, coimage and cokernel of $u$ are denoted respectively by Ker $u$, Im $u$, Coim $u$, Coker $u$.