Module theory MOC

Quotient module

Given a Module and a Submodule , the quotient module is the quotient group with a natural -action: module

for any and .

More explicitly

is just with all elements of collapsed to zero. More formally, using the congruence relation

we have with .

We thus have the short exact sequence in

Universal property

The quotient module with the canonical projection is characterised up to unique isomorphism by the universal property:

. If is a module and is a morphism with , then there exists a unique morphism so that .

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