# Quotient module

In algebra, given a module and a submodule, one can construct their **quotient module**.^{[1]}^{[2]} 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*] = { *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, [*a*] + [*b*] = [*a*+*b*], and *r*·[*a*] = [*r*·*a*], for all *a*,*b* in *A* and *r* in *R*.

## ExamplesEdit

Consider the ring **R** of real numbers, and the **R**-module *A* = **R**[*X*], that is the polynomial ring with real coefficients. Consider the submodule

*B*= (*X*^{2}+ 1)**R**[*X*]

of *A*, that is, the submodule of all polynomials divisible by *X*^{2}+1. It follows that the equivalence relation determined by this module will be

*P*(*X*) ~*Q*(*X*) if and only if*P*(*X*) and*Q*(*X*) give the same remainder when divided by*X*^{2}+ 1.

Therefore, in the quotient module *A*/*B*, *X*^{2} + 1 is the same as 0; so one can view *A*/*B* as obtained from **R**[*X*] by setting *X*^{2} + 1 = 0. This quotient module is isomorphic to the complex numbers, viewed as a module over the real numbers **R**.

## See alsoEdit

## ReferencesEdit

**^**Dummit, David S.; Foote, Richard M. (2004).*Abstract Algebra*(3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.**^**Lang, Serge (2002).*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.