- Quotient module
In
abstract algebra , a branch ofmathematics , given a module and asubmodule , one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring ofinteger smodulo an integer "n", seemodular arithmetic . It is the same construction used forquotient group s andquotient ring s.Given a module "A" over a ring "R", and a
submodule "B" of "A", thequotient space "A"/"B" is defined by theequivalence 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".Examples
Consider the ring R of
real number s, and the R-module "A" = R ["X"] , that is thepolynomial 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" one will have "X"2 + 1 be the same as 0, and such, one can view "A"/"B" as obtained from R ["X"] by setting "X"2 + 1 = 0. It is clear that this quotient module will be
isomorphic to thecomplex number s, viewed as a module over the real numbers R.ee also
*
quotient group
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