In algebra, given a module and a submodule, one can construct their quotient module.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> This construction, described below, is very similar to that of a quotient vector space.<ref>Template:Cite book</ref> 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 (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 Template:Mvar over a ring Template:Mvar, and a submodule Template:Mvar of Template:Mvar, the quotient space Template:Math is defined by the equivalence relation
- <math>a \sim b</math> if and only if <math>b - a \in B,</math>
for any Template:Mvar in Template:Mvar.<ref>Template:Harvnb</ref> The elements of Template:Math are the equivalence classes <math>[a] = a+B = \{a+b:b \in B\}.</math> The function <math>\pi: A \to A/B</math> sending Template:Mvar in Template:Mvar to its equivalence class Template:Math is called the quotient map or the projection map, and is a module homomorphism.
The addition operation on Template:Math 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 Template:Math by elements of Template:Mvar is defined similarly. Note that it has to be shown that these operations are well-defined. Then Template:Math becomes itself an Template:Mvar-module, called the quotient module. In symbols, for all Template:Mvar in Template:Mvar and Template:Mvar in Template:Mvar:
- <math>\begin{align}
& (a+B)+(b+B) := (a+b)+B, \\ & r \cdot (a+B) := (r \cdot a)+B. \end{align}</math>
ExamplesEdit
Consider the polynomial ring, Template:Tmath with real coefficients, and the Template:Tmath-module <math>A=\R[X],</math> . Consider the submodule
- <math>B = (X^2+1) \R[X]</math>
of Template:Mvar, that is, the submodule of all polynomials divisible by Template:Math. It follows that the equivalence relation determined by this module will be
- Template:Math if and only if Template:Math and Template:Math give the same remainder when divided by Template:Math.
Therefore, in the quotient module Template:Math, Template:Math is the same as 0; so one can view Template:Math as obtained from Template:Tmath by setting Template:Math. This quotient module is isomorphic to the complex numbers, viewed as a module over the real numbers Template:Tmath