Editing Quotient module

{{short description|Algebraic construction}}

In [[algebra]], given a [[module (mathematics)|module]] and a [[submodule]], one can construct their '''quotient module'''.<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref>  This construction, described below, is very similar to that of a [[quotient vector space]].<ref>{{cite book |last=Roman |first=Steven |date=2008 |title=Advanced linear algebra |edition=3rd |location=New York |publisher=Springer Science + Business Media |isbn=978-0-387-72828-5 |page=117}}</ref> It differs from analogous quotient constructions of [[Ring (mathematics)|rings]] and [[Group (mathematics)|groups]] by the fact that in the latter cases, the [[Linear subspace|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 (ring theory)|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 {{mvar|A}} over a ring {{mvar|R}}, and a submodule {{mvar|B}} of {{mvar|A}}, the [[Quotient space (topology)|quotient space]] {{math|''A''/''B''}} is defined by the [[equivalence relation]]

: <math>a \sim b</math> [[if and only if]] <math>b - a \in B,</math>

for any {{mvar|a, b}} in {{mvar|A}}.<ref>{{harvnb|Roman|2008|loc=p. 118 Theorem 4.7}}</ref> The elements of {{math|''A''/''B''}} are the [[equivalence class]]es <math>[a] = a+B = \{a+b:b \in B\}.</math> The [[Function (mathematics)|function]] <math>\pi: A \to A/B</math> sending {{mvar|a}} in {{mvar|A}} to its equivalence class {{math|''a'' + ''B''}} is called the ''[[Quotient map (topology)|quotient map]]'' or the ''projection map'', and is a [[module homomorphism]].

The [[addition]] [[Operation (mathematics)|operation]] on {{math|''A''/''B''}} is defined for two equivalence classes as the equivalence class of the sum of two [[Representative (mathematics)|representatives]] from these classes; and [[scalar multiplication]] of elements of {{math|''A''/''B''}} by elements of {{mvar|R}} is defined similarly.  Note that it has to be shown that these operations are [[Well-defined expression|well-defined]].  Then {{math|''A''/''B''}} becomes itself an {{mvar|R}}-module, called the ''quotient module''.  In symbols, for all {{mvar|a, b}} in {{mvar|A}} and {{mvar|r}} in {{mvar|R}}:
:<math>\begin{align}
& (a+B)+(b+B) := (a+b)+B, \\
& r \cdot (a+B) := (r \cdot a)+B.
\end{align}</math>

==Examples==

Consider the [[polynomial ring]], {{tmath|\R[X]}} with real [[Coefficient|coefficients]], and the {{tmath|\R[X]}}-module <math>A=\R[X],</math> . Consider the submodule

:<math>B = (X^2+1) \R[X]</math>

of {{mvar|A}}, that is, the submodule of all polynomials divisible by {{math|''X''{{sup| 2}} + 1}}. It follows that the equivalence relation determined by this module will be

:{{math|''P''(''X'') ~ ''Q''(''X'')}} if and only if {{math|''P''(''X'')}} and {{math|''Q''(''X'')}} give the same remainder when divided by {{math|''X''{{sup| 2}} + 1}}.

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

==See also==
* [[Quotient group]]
* [[Quotient ring]]
* [[Quotient (universal algebra)]]

==References==
{{reflist}}

[[Category:Module theory]]
[[Category:Quotient objects|Module]]