Quotient module

In abstract algebra, given a module and a submodule, one can construct their quotient module.^[1][2] This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic. It is the same construction used for quotient groups and quotient rings.

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 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 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 also

• quotient group

References