Ideal (ring Theory) - Ideals and Congruence Relations

Ideals and Congruence Relations

There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring:

Given an ideal I of a ring R, let x ~ y if xyI. Then ~ is a congruence relation on R.

Conversely, given a congruence relation ~ on R, let I = {x : x ~ 0}. Then I is an ideal of R.

Read more about this topic:  Ideal (ring Theory)

Famous quotes containing the words ideals, congruence and/or relations:

    You can tell the ideals of a nation by its advertisements.
    Norman Douglas (1868–1952)

    As for butterflies, I can hardly conceive
    of one’s attending upon you; but to question
    the congruence of the complement is vain, if it exists.
    Marianne Moore (1887–1972)

    When any one of our relations was found to be a person of a very bad character, a troublesome guest, or one we desired to get rid of, upon his leaving my house I ever took care to lend him a riding-coat, or a pair of boots, or sometimes a horse of small value, and I always had the satisfaction of finding he never came back to return them.
    Oliver Goldsmith (1728–1774)