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∈A for all elements r of R and a of A.
Examples
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1.
The subsets {0} and T are always submodules of the module T.
-
2.
The set {t∈T:rt=t∀r∈R} of all invariant elements of T is a submodule of T.
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3.
If X⊆T and 𝔞 is a left ideal
of R, then the set
𝔞X:= is a submodule of . Especially, is called the submodule generated by the subset ; then the elements of are generators
of this submodule.
There are some operations on submodules. Given the submodules and of , the sum and the intersection
are submodules of .
The notion of sum may be extended for any family of submodules: the sum of submodules consists of all finite sums where every belongs to one of those submodules. The sum of submodules as well as the intersection are submodules of . The submodule is the intersection of all submodules containing the subset .
If is a ring and is a subring of , then is an -module; then one can consider the product and the quotient of the left -submodules and of :
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•
-
•
Also these are left -submodules of .
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 |