# 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. 1.

The subsets $\{0\}$ and $T$ are always submodules of the module $T$.

2. 2.

The set  $\{t\in T:\,\,\,rt=t\,\,\,\forall r\in R\}$  of all invariant elements of $T$ is a submodule of $T$.

3. 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 $X$; 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 $RX$ 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 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