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    division algorithm

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    Let R={a+b(sqrt(2)) | a, bâ?? Z}.
    (a): Show that R is a ring.
    (b): Let M = {a+b(sqrt(2)) â?? R | 5|a and 5|b}. Show that M is a maximal ideal of R.

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    https://brainmass.com/math/basic-algebra/modern-algebra-ring-357108

    Solution Preview

    Let

    (a) Show that R is a ring.
    (b) Let Show that M is a maximal ideal of R.

    Proof:

    (a) To show that R is a ring, we'll show that R is a subring of the field R of real numbers.

    Since it follows that
    Let and where Then
    where and
    So
    Finally,

    where are integers.
    So
    Therefore, R is a subring of the field R of real numbers.

    (b) Let To show that M is an ideal, let and Then we have

    where are integers such that and
    But then we have and
    So and ...

    Solution Summary

    A division algorithm is applied.

    $2.19

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