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 $\U0001d51e$ is a left ideal^{} of $R$, then the set
$$\U0001d51eX:=\{\text{finite}\sum _{\nu}{a}_{\nu}{x}_{\nu}:{a}_{\nu}\in \U0001d51e,{x}_{\nu}\in X\forall \nu \}$$ 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 $A+B:=\{a+b\in T:a\in A\wedge b\in B\}$ 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 $T$:

•
$AB:=\{\text{finite}{\sum}_{\nu}{a}_{\nu}{b}_{\nu}:{a}_{\nu}\in A,{b}_{\nu}\in B\forall \nu \}$

•
$[A:B]:=\{t\in T:tB\subseteq A\}$
Also these are left $R$submodules of $T$.
Title  submodule 
Canonical name  Submodule 
Date of creation  20130322 15:15:26 
Last modified on  20130322 15:15:26 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  19 
Author  PrimeFan (13766) 
Entry type  Definition 
Classification  msc 2000 
Classification  msc 1600 
Classification  msc 1300 
Related topic  SumOfIdeals 
Related topic  QuotientOfIdeals 
Defines  Rsubmodule 
Defines  generated submodule 
Defines  generator 
Defines  sum of submodules 
Defines  product submodule 
Defines  quotient of submodules 