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Viewing Version
9
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'submodule'
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| Title of object: |
submodule |
| Canonical Name: |
Submodule |
| Type: |
Definition |
| Created on: |
2005-05-11 19:10:44 |
| Modified on: |
2005-07-11 17:11:00 |
| Classification: |
msc:13-00, msc:16-00, msc:20-00 |
| Defines: |
R-submodule, generated submodule, sum of submodules, quotient of submodules |
Preamble:
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\newtheorem*{thmplain}{Theorem} |
Content:
Let $R$ be a ring and $T$ a left $R$-module. \,A subset $A$ of $T$ is called a ({\em left}) {\em submodule} of $T$, if $A$ is a left $R$-module.
\textbf{Examples}
\begin{enumerate}
\item The subsets $\{0\}$ and $T$ are always submodules of the module $T$.
\item The set \,$\{t\in T:\,\,\,rt = t\,\,\,\forall r\in R\}$\, of all invariant elements of $T$ is a submodule of $T$.
\item 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 {\em generated} by the subset $X$.
\end{enumerate}
There are some operations on submodules.\, Let $A$ and $B$ be submodules of $T$. \,Then the {\em 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 $RX$ is the intersection of all submodules containing the subset $X$.
If $T$ is a ring and $R$ is a subring of $T$, then one can consider the {\em product} and the {\em quotient} of the left submodules $A$ and $B$ of $T$:
\begin{itemize}
\item $AB := \{\mbox{finite}\sum_\nu a_\nu b_\nu:
\,\,\,a_\nu\in A,\,\,b_\nu\in X\,\,\forall\nu\}$
\item $[A:B] := \{t\in T:\,\, tB\subseteq A\}$
\end{itemize}
Also these are left submodules of $T$. |
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