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:
“Our chaotic economic situation has convinced so many of our young people that there is no room for them. They become uncertain and restless and morbid; they grab at false promises, embrace false gods and judge things by treacherous values. Their insecurity makes them believe that tomorrow doesn’t matter and the ineffectualness of their lives makes them deny the ideals which we of an older generation acknowledged.”
—Hortense Odlum (1892–?)
“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)
“The principle that human nature, in its psychological aspects, is nothing more than a product of history and given social relations removes all barriers to coercion and manipulation by the powerful.”
—Noam Chomsky (b. 1928)