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A map that kills a submodule factors uniquely through the quotient.
Universal property of quotient modules: Let be an -module, let be a submodule, and let be the quotient map. For any -module and any homomorphism such that , there exists a unique homomorphism with .
This is the defining mapping property of the quotient module, and it expresses quotients as the universal way to force a submodule to lie in the kernel of a module homomorphism.
Let be a left -module and let be a submodule. Define an equivalence relation on by iff . The quotient module is the quotient set of equivalence classes, written , with operations
These operations are well-defined precisely because is closed under subtraction and scalar multiplication.
The construction is characterized by the universal property of the quotient module: maps out of that kill factor uniquely through .
Let be a module homomorphism. The kernel of is
It is a submodule, as recorded in kernels are submodules.
Kernels measure injectivity: is injective iff . They also define the notion of exactness (see exact sequences, where kernels match images).