The ideal quotient satisfies the following properties:

• ${\displaystyle (I:J)=\mathrm {Ann} _{R}((J+I)/I)}$  as ${\displaystyle R}$ -modules, where ${\displaystyle \mathrm {Ann} _{R}(M)}$  denotes the annihilator of ${\displaystyle M}$  as an ${\displaystyle R}$ -module.
• ${\displaystyle J\subseteq I\Leftrightarrow (I:J)=R}$  (in particular, ${\displaystyle (I:I)=(R:I)=(I:0)=R}$ )
• ${\displaystyle (I:R)=I}$ 
• ${\displaystyle (I:(JK))=((I:J):K)}$ 
• ${\displaystyle (I:(J+K))=(I:J)\cap (I:K)}$ 
• ${\displaystyle ((I\cap J):K)=(I:K)\cap (J:K)}$ 
• ${\displaystyle (I:(r))={\frac {1}{r}}(I\cap (r))}$  (as long as R is an integral domain)

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, ..., x_n], then

${\displaystyle I:J=(I:(g_{1}))\cap (I:(g_{2}))=\left({\frac {1}{g_{1}}}(I\cap (g_{1}))\right)\cap \left({\frac {1}{g_{2}}}(I\cap (g_{2}))\right)}$ 

Then elimination theory can be used to calculate the intersection of I with (g₁) and (g₂):

${\displaystyle I\cap (g_{1})=tI+(1-t)(g_{1})\cap k[x_{1},\dots ,x_{n}],\quad I\cap (g_{2})=tI+(1-t)(g_{2})\cap k[x_{1},\dots ,x_{n}]}$ 

Calculate a Gröbner basis for ${\displaystyle tI+(1-t)(g_{1})}$  with respect to lexicographic order. Then the basis functions which have no t in them generate ${\displaystyle I\cap (g_{1})}$ .

The ideal quotient corresponds to set difference in algebraic geometry.^[1] More precisely,

• If W is an affine variety (not necessarily irreducible) and V is a subset of the affine space (not necessarily a variety), then

${\displaystyle I(V):I(W)=I(V\setminus W)}$ 

where ${\displaystyle I(\bullet )}$  denotes the taking of the ideal associated to a subset.

• If I and J are ideals in k[x₁, ..., x_n], with k an algebraically closed field and I radical then

${\displaystyle Z(I:J)=\mathrm {cl} (Z(I)\setminus Z(J))}$ 

where ${\displaystyle \mathrm {cl} (\bullet )}$  denotes the Zariski closure, and ${\displaystyle Z(\bullet )}$  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:

${\displaystyle Z(I:J^{\infty })=\mathrm {cl} (Z(I)\setminus Z(J))}$ 

where ${\displaystyle (I:J^{\infty })=\cup _{n\geq 1}(I:J^{n})}$ .

• In ${\displaystyle \mathbb {Z} }$  we have ${\displaystyle ((6):(2))=(3)}$ .
• In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal ${\displaystyle I}$  of an integral domain ${\displaystyle R}$  is given by the ideal quotient ${\displaystyle ((1):I)=I^{-1}}$ .
• One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let ${\displaystyle I=(xyz),J=(xy)}$  in ${\displaystyle \mathbb {C} [x,y,z]}$  be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in ${\displaystyle \mathbb {A} _{\mathbb {C} }^{3}}$ . Then, the ideal quotient ${\displaystyle (I:J)=(z)}$  is the ideal of the z-plane in ${\displaystyle \mathbb {A} _{\mathbb {C} }^{3}}$ . 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 ${\displaystyle ((x^{4}y^{3}):(x^{2}y^{2}))=(x^{2}y)}$ , 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 ${\displaystyle I\subset R[x_{0},\ldots ,x_{n}]}$  the saturation of ${\displaystyle I}$  is defined as the ideal quotient ${\displaystyle (I:{\mathfrak {m}}^{\infty })=\cup _{i\geq 1}(I:{\mathfrak {m}}^{i})}$  where ${\displaystyle {\mathfrak {m}}=(x_{0},\ldots ,x_{n})\subset R[x_{0},\ldots ,x_{n}]}$ . It is a theorem that the set of saturated ideals of ${\displaystyle R[x_{0},\ldots ,x_{n}]}$  contained in ${\displaystyle {\mathfrak {m}}}$  is in bijection with the set of projective subschemes in ${\displaystyle \mathbb {P} _{R}^{n}}$ .^[2] This shows us that ${\displaystyle (x^{4}+y^{4}+z^{4}){\mathfrak {m}}^{k}}$  defines the same projective curve as ${\displaystyle (x^{4}+y^{4}+z^{4})}$  in ${\displaystyle \mathbb {P} _{\mathbb {C} }^{2}}$ .

• Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012)

• M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.