In algebra, given a module and a submodule, one can construct their quotient module.[1] [2] This construction, described below, is very similar to that of a quotient vector space.[3] It differs from analogous quotient constructions of rings and groups by the fact that in these cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring, and a quotient group is the quotient of a group by a normal subgroup, not by a general subgroup).
Given a module over a ring, and a submodule of, the quotient space is defined by the equivalence relation
a\simb
b-a\inB,
for any in . The elements of are the equivalence classes
[a]=a+B=\{a+b:b\inB\}.
\pi:A\toA/B
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 :
\begin{align} &(a+B)+(b+B):=(a+b)+B,\\ &r ⋅ (a+B):=(r ⋅ a)+B. \end{align}
Consider the polynomial ring, with real coefficients, and the -module
A=\R[X],
B=(X2+1)\R[X]
of, that is, the submodule of all polynomials divisible by . It follows that the equivalence relation determined by this module will be
if and only if and give the same remainder when divided by .
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
. Serge Lang . Algebra . . . 2002 . 0-387-95385-X.