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In algebra, given a module and a submodule, one can construct their quotient module.[1][2] This construction, described below, is very similar to that of a quotient vector space. It differs from analogous constructions of quotient groups and quotient rings by the fact that in these cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space (that is, the quotient of a group by a subgroup is not always a group, and a quotient ring is the quotient of a ring by an ideal, not a subring).

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 ba 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 = (X2 + 1) R[X]

of A, that is, the submodule of all polynomials divisible by X2+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 X2 + 1.

Therefore, in the quotient module A/B, X2 + 1 is the same as 0; so one can view A/B as obtained from R[X] by setting X2 + 1 = 0. This quotient module is isomorphic to the complex numbers, viewed as a module over the real numbers R.

See alsoEdit

ReferencesEdit

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