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 $𝔞$ is a left ideal of $R$, then the set

   $$𝔞X:=\{\text{finite}\sum _{\nu }{a}_{\nu }{x}_{\nu }:a_{\nu }\in 𝔞,x_{\nu }\in X\forall \nu \}$$

   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  $A+B:=\{a+b\in T:a\in A\wedge b\in B\}$  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	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