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For a homomorphism f, the induced map M/ker(f) → im(f) is an isomorphism.
Quotient by kernel is isomorphic to image: Let be an -module homomorphism. Then the induced map
is a well-defined isomorphism of -modules.
This is the usual “image–kernel” form of the first isomorphism theorem for modules, relating quotient modules to kernels and images.
First isomorphism theorem (modules): Let be a module homomorphism. Then the induced map
is an isomorphism of -modules. Here is the kernel and is the image, and is a quotient module.
This theorem identifies the “effective domain” of a map with its image and is the basic mechanism behind many results in the theory of exact sequences; it is the module analogue of the first isomorphism theorem for rings
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).
Let be a module homomorphism. The image of is
It is a submodule of ; see kernel/image are submodules for the standard closure argument.
Images measure surjectivity: is surjective iff . Together with kernels, images define exactness via the condition for consecutive maps (see exactness via kernels and images