Given a ring $R$ and a left $R$ -module $T$ , a subset $A$ of $T$ is called a (left) submodule of $T$ , if $(A,\,+)$ is a subgroup of $(M,\,+)$ and $ra \in A$ for all elements $r$ of $R$ and $a$ of $A$ .

Examples

1. The subsets $\{0\}$ and $T$ are always submodules of the module $T$ .
2. The set $\{t\in T:\,\,\,rt = t\,\,\,\forall r\in R\}$ of all invariant elements of $T$ is a submodule of $T$ .
3. If $X \subseteq T$ and $\mathfrak{a}$ is a left ideal of $R$ , then the set $$\mathfrak{a}X := \{\mbox{finite}\sum_\nu a_\nu x_\nu: \,\,\,a_\nu\in\mathfrak{a},\,\,x_\nu\in X\,\,\forall\nu\}$$ is a submodule of $T$ . Especially, $RX$ is called the submodule generated by the subset $X$ .

There are some operations on submodules. Given the submodules $A$ and $B$ of $T$ , the sum $A + B := \{a + b\in T:\,\,a\in A \,\land\, b\in B\}$ and the intersection $A\cap B$ are submodules of $T$ .

The notion of sum may be extended for any family $\{A_j:\,\,j\in J\}$ of submodules: the sum $\sum_{j\in J}A_j$ of submodules consists of all finite sums $\sum_j a_j$ where every $a_j$ belongs to one $A_j$ of those submodules. The sum of submodules as well as the intersection $\bigcap_{j\in J}A_j$ are submodules of $T$ . The submodule $RX$ is the intersection of all submodules containing the subset $X$ .

If $T$ is a ring and $R$ is a subring of $T$ , then $T$ is an $R$ -module; then one can consider the product and the quotient of the left $R$ -submodules $A$ and $B$ of $T$ :

• $AB := \{\mbox{finite}\sum_\nu a_\nu b_\nu: \,\,\,a_\nu\in A,\,\,b_\nu\in B\,\,\forall\nu\}$
• $[A:B] := \{t\in T:\,\, tB\subseteq A\}$