fundamental theorem of module homomorphisms
1. Proposition
Let 
be a ring, 
a 
-module 
a submodule.
Suppose
is the canonical projection to the quotient module
Then for each module-homomorphism
such that 
, then there exists a unique module-homomorphism 
such that
2. Proof
2.1. uniqueness
Suppose we have 
satisfying the condition.
then we get
Since 
is surjective, by surjective map as epimorphism we conclude, that is an epimorphism.
Hence it follows, that
2.2. existence
Let
2.2.1. welldefined map
Let 
.
Then there exists an 
such that 
.
Therefore,
2.2.2. module homomorphism
2.2.2.1. additive
2.2.2.2. multiplication
2.2.3. commutes
follows from construction