Quotient Module

In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic. It is the same construction used for quotient groups and quotient rings.

Given a module A over a ring R, and a submodule B of A, the quotient space A/B is defined by the equivalence relation

a ~ b if and only if b − a is in B,

for any a and b in A. The elements of A/B are the equivalence classes = { a + b : b in B }.

The addition operation on A/B is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and in the same way for multiplication by elements of R. In this way A/B becomes itself a module over R, called the quotient module. In symbols, + =, and r· =, for all a,b in A and r in R.