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[parent] submodule (Definition)

Let $ R$ be a ring and $ T$ a left $ R$-module. A subset $ A$ of $ T$ is called a (left) submodule of $ T$, if $ A$ is a left $ R$-module.

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
    $\displaystyle \mathfrak{a}X := \{$finite$\displaystyle \sum_\nu a_\nu x_\nu: \,\,\,a_\nu\in\mathfrak{a},\,\,x_\nu\in X\,\,\forall\nu\}$
    is a submodule of $ T$. Especially, $ RX$ is called the submodule generated by the subset $ X$.

There are some operations on submodules. Let $ A$ and $ B$ be submodules of $ T$. Then 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 product and the quotient of the left $ R$-submodules $ A$ and $ B$ of $ T$:

  • $ AB := \{$finite$ \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$.



"submodule" is owned by Mravinci. [ owner history (1) ]
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See Also: sum of ideals, quotient of ideals

Also defines:  R-submodule, generated submodule, sum of submodules, product submodule, quotient of submodules

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Cross-references: quotient, product, subring, finite, intersection, sum, operations, left ideal, invariant, module, subset, ring
There are 50 references to this entry.

This is version 12 of submodule, born on 2005-05-11, modified 2007-04-16.
Object id is 7040, canonical name is Submodule.
Accessed 4743 times total.

Classification:
AMS MSC: 13-00 (Commutative rings and algebras :: General reference works )
 16-00 (Associative rings and algebras :: General reference works )
 20-00 (Group theory and generalizations :: General reference works )
Pending Errata and Addenda
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