In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set
![{\displaystyle (I:J)=\{r\in R\mid rJ\subseteq I\}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a6464d6873ab5e97bc93f1453c7415ac56f67fd9)
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Not to be confused with the
quotient of a ring by an ideal.
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 (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.
The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = (f1, f2, f3) and J = (g1, g2) are ideals in k[x1, ..., xn], then
![{\displaystyle I:J=(I:(g_{1}))\cap (I:(g_{2}))=\left({\frac {1}{g_{1}}}(I\cap (g_{1}))\right)\cap \left({\frac {1}{g_{2}}}(I\cap (g_{2}))\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/862693a2e41705dbf70e7e679a1b968b314db6d7)
Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):
![{\displaystyle I\cap (g_{1})=tI+(1-t)(g_{1})\cap k[x_{1},\dots ,x_{n}],\quad I\cap (g_{2})=tI+(1-t)(g_{2})\cap k[x_{1},\dots ,x_{n}]}](//wikimedia.org/api/rest_v1/media/math/render/svg/d383a1c89a40fa4a34930e38844e529eec480c7c)
Calculate a Gröbner basis for
with respect to lexicographic order. Then the basis functions which have no t in them generate
.
- In
, ![{\displaystyle ((6):(2))=(3)}](//wikimedia.org/api/rest_v1/media/math/render/svg/b3c9ad98a0ed5b082801171c63f9cdd9d5b58889)
- In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal
of an integral domain
is given by the ideal quotient
.
- One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let
in
be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in
. Then, the ideal quotient
is the ideal of the z-plane in
. This shows how the ideal quotient can be used to "delete" irreducible subschemes.
- A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient
, showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure.
- We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal
the saturation of
is defined as the ideal quotient
where
. It is a theorem that the set of saturated ideals of
contained in
is in bijection with the set of projective subschemes in
.[2] This shows us that
defines the same projective curve as
in
.
David Cox; John Little; Donal O'Shea (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer. ISBN 0-387-94680-2., p.195
- M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.