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 x − y ∈ I. 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:
“With the breakdown of the traditional institutions which convey values, more of the burdens and responsibility for transmitting values fall upon parental shoulders, and it is getting harder all the time both to embody the virtues we hope to teach our children and to find for ourselves the ideals and values that will give our own lives purpose and direction.”
—Neil Kurshan (20th century)
“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 one walks, one is brought into touch first of all with the essential relations between one’s physical powers and the character of the country; one is compelled to see it as its natives do. Then every man one meets is an individual. One is no longer regarded by the whole population as an unapproachable and uninteresting animal to be cheated and robbed.”
—Aleister Crowley (1875–1947)