Ideal quotient

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 formula_2 if and only if formula_3. The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below).
("I" : "J") is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.
Properties.
The ideal quotient satisfies the following properties:
Calculating the quotient.
The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if "I" = ("f"1, "f"2, "f"3) and "J" = ("g"1, "g"2) are ideals in "k"["x"1, ..., "x""n"], then
Then elimination theory can be used to calculate the intersection of "I" with ("g"1) and ("g"2):
Calculate a Gröbner basis for "tI" + (1-"t")("g"1) with respect to lexicographic order. Then the basis functions which have no "t" in them generate formula_16.
Geometric interpretation.
The ideal quotient corresponds to set difference in algebraic geometry. More precisely,
where formula_18 denotes the taking of the ideal associated to a subset.
where formula_20 denotes the Zariski closure, and formula_21 denotes the taking of the variety defined by an ideal.
If "I" is not radical, then the same property holds if we saturate the ideal "J":
where formula_23.