Examples
Consider the ring R of real numbers, and the R-module A = R, that is the polynomial ring with real coefficients. Consider the submodule
- B = (X2 + 1) R
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 by setting X2 + 1 = 0. This quotient module is isomorphic to the complex numbers, viewed as a module over the real numbers R.
Read more about this topic: Quotient Module
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