Editing Quotient module (section)

==Examples==

Consider the [[polynomial ring]], {{tmath|\R[X]}} with real [[Coefficient|coefficients]], and the {{tmath|\R[X]}}-module <math>A=\R[X],</math> . Consider the submodule

:<math>B = (X^2+1) \R[X]</math>

of {{mvar|A}}, that is, the submodule of all polynomials divisible by {{math|''X''{{sup| 2}} + 1}}. It follows that the equivalence relation determined by this module will be

:{{math|''P''(''X'') ~ ''Q''(''X'')}} if and only if {{math|''P''(''X'')}} and {{math|''Q''(''X'')}} give the same remainder when divided by {{math|''X''{{sup| 2}} + 1}}.

Therefore, in the quotient module {{math|''A''/''B''}}, {{math|''X''{{sup| 2}} + 1}} is the same as 0; so one can view {{math|''A''/''B''}} as obtained from  {{tmath|\R[X]}} by setting {{math|1=''X''{{sup| 2}} + 1 = 0}}. This quotient module is [[isomorphic]] to the [[complex number]]s, viewed as a module over the real numbers {{tmath|\R.}}