In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set
Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry.
(I : J) is sometimes referred to as a colon ideal because of the notation. There is an unrelated notion of the inverse of an ideal, known as a fractional ideal which is defined for Dedekind rings.
Read more about Ideal Quotient: Properties, Calculating The Quotient, Geometric Interpretation
Famous quotes containing the word ideal:
“But I must needs take my petulance, contrasting it with my accustomed morning hopefulness, as a sign of the ageing of appetite, of a decay in the very capacity of enjoyment. We need some imaginative stimulus, some not impossible ideal which may shape vague hope, and transform it into effective desire, to carry us year after year, without disgust, through the routine- work which is so large a part of life.”
—Walter Pater (1839–1894)