Quotient module

 ( Module Theory ) 

In algebra, given a module and a submodule, one can construct their quotient module.

Serge Lang (2025). 038795385X, Springer. 038795385X

This construction, described below, is very similar to that of a quotient vector space.

Steven Roman (2025). 9780387728285, Springer Science + Business Media. ISBN 9780387728285

It differs from analogous quotient constructions of rings and groups by the fact that in the latter cases, the Linear 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

$a\; \backslash sim\; b$ if and only if $b\; -\; a\; \backslash in\; B,$

for any in . The elements of are the equivalence classes $a\; =\; a+B\; =\; \backslash \{a+b:b\; \backslash in\; B\backslash \}.$ The function $\backslash pi:\; A\; \backslash to\; A/B$ sending in to its equivalence class is called the quotient map or the projection map, and is a module homomorphism.

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 :

$\backslash begin\{align\}$

& (a+B)+(b+B) := (a+b)+B, \\ & r \cdot (a+B) := (r \cdot a)+B. \end{align}

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Examples Consider the polynomial ring, with real Coefficient, and the -module $A=\backslash RX,$ . Consider the submodule

$B\; =\; (X^2+1)\; \backslash RX$

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 , viewed as a module over the real numbers

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See also

• Quotient group
• Quotient ring
• Quotient (universal algebra)

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