Quotient module

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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:

• 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

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See also

• Quotient group
• Quotient ring
• Quotient