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.
Other articles related to "ideal quotient, ideal, ideals":
... The ideal quotient corresponds to set difference in algebraic geometry ... I(V W), where I denotes the taking of the ideal associated to a subset ... If I and J are ideals in k, then Z(I J) = cl(Z(I) Z(J)) where "cl" denotes the Zariski closure, and Z denotes the taking of the variety defined by the ideal I ...
Famous quotes containing the word ideal:
“One who does not know how to discover the pathway to his ideal lives more frivolously and impudently than the man without an ideal.”
—Friedrich Nietzsche (18441900)