Quotient module
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by the fact that in the latter cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space.
Given a module over a ring, and a submodule of, the Quotient [space (topology)|quotient space] is defined by the equivalence relation
for any in. The elements of are the equivalence classes The function sending in to its equivalence class is called the quotient map or the projection map, and is a module homomorphism.
The addition operation on is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and scalar multiplication of elements of by elements of is defined similarly. Note that it has to be shown that these operations are well-defined. Then becomes itself an -module, called the quotient module. In symbols, for all in and in :
Examples
Consider the polynomial ring, with real coefficients, and the -module . Consider the submoduleof, that is, the submodule of all polynomials divisible by. It follows that the equivalence relation determined by this module will be
Therefore, in the quotient module, is the same as 0; so one can view as obtained from by setting. This quotient module is isomorphic to the complex numbers, viewed as a module over the real numbers