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For submodules A,B ≤ M, one has (A+B)/B ≅ A/(A∩B).
Second isomorphism theorem (modules): Let be an -module and let be submodules of . Then there is a natural isomorphism of -modules
where is the intersection and each quotient is a quotient module.
This isomorphism is obtained by restricting the quotient map to , and it is a standard application of the first isomorphism theorem.
An intersection is the set of elements common to every set in a given collection. For sets ,
More generally, for an indexed family of sets ,
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 .
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