Quotient module

From Maths
Revision as of 20:00, 23 October 2016 by Alec (Talk | contribs) (Created page with "{{Stub page|grade=A*|msg=Improve and get another source! Demote when this page has been written and there is a map.}} __TOC__ ==Definition== Let {{M|(R,*,+,0)}} be a ring...")

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Improve and get another source! Demote when this page has been written and there is a map.

Definition

Let [ilmath](R,*,+,0)[/ilmath] be a ring (with or without unity) and let [ilmath]M[/ilmath] a (left) [ilmath]R[/ilmath]-module. Let [ilmath]A\subseteq M[/ilmath] be a submodule of [ilmath]M[/ilmath]. Then[1]:

  • The quotient group [ilmath]\frac{M}{A} [/ilmath] is actually a (left) module too with the operations:
    1. (ADDITION) - given by the quotient group part
      • [ilmath]+:\frac{M}{A}\times\frac{M}{A}\rightarrow\frac{M}{A} [/ilmath] by [ilmath]+:([x],[y])\mapsto [x+y][/ilmath]
    2. Multiplication/module action: [ilmath]\cdot:R\times\frac{M}{A}\rightarrow\frac{M}{A} [/ilmath] by [ilmath]\cdot:(r,[x])\mapsto [rx][/ilmath]

Furthermore, if [ilmath]M[/ilmath] is a unital module then so is [ilmath]\frac{M}{A} [/ilmath]


With this we get a canonical projection, [ilmath]\pi:M\rightarrow\frac{M}{A} [/ilmath] that is a module homomorphism:

  • [ilmath]\pi:x\mapsto [x][/ilmath]

and the kernel is [ilmath]A[/ilmath].

Characteristic property of the quotient module

Characteristic property of the quotient module/Statement

Proof

Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Very important for everything

See also

References

  1. Abstract Algebra - Pierre Antoine Grillet