submodule

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, $R X$ is called the submodule generated by the subset $X$ ; then the elements of $X$ are generators of this submodule.

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 $R X$ 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 $T$ :

•

$AB:=\{\mbox{finite}\sum_{\nu}a_{\nu}b_{\nu}:\,\,\,a_{\nu}\in A,\,\,b_{\nu}\in B% \,\,\forall\nu\}$
•

$[A:B]:=\{t\in T:\,\,tB\subseteq A\}$

Also these are left $R$ -submodules of $T$ .

Title	submodule
Canonical name	Submodule
Date of creation	2013-03-22 15:15:26
Last modified on	2013-03-22 15:15:26
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	19
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 20-00
Classification	msc 16-00
Classification	msc 13-00
Related topic	SumOfIdeals
Related topic	QuotientOfIdeals
Defines	R-submodule
Defines	generated submodule
Defines	generator
Defines	sum of submodules
Defines	product submodule
Defines	quotient of submodules