# annihilator -- the annihilator ideal

## Synopsis

• Usage:
ann M
annihilator M
• Inputs:
• Optional inputs:
• Strategy => , default value null, either Quotient or Intersection
• Outputs:
• an ideal, the annihilator ideal $\mathrm{ann}(M) = \{ f \in R | fM = 0 \}$ where $R$ is the ring of $M$

## Description

You may use ann as a synonym for annihilator.

As an example, we compute the annihilator of the canonical module of the rational quartic curve.

 i1 : R = QQ[a..d]; i2 : J = monomialCurveIdeal(R,{1,3,4}) 3 2 2 2 3 2 o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c) o2 : Ideal of R i3 : M = Ext^2(R^1/J, R) o3 = cokernel {-3} | c a 0 b 0 | {-3} | -d -b c 0 a | {-3} | 0 0 d c b | 3 o3 : R-module, quotient of R i4 : annihilator M 3 2 2 2 3 2 o4 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c) o4 : Ideal of R

For another example, we compute the annihilator of an element in a quotient ring.

 i5 : A = R/(a*b, a*c, a*d) o5 = A o5 : QuotientRing i6 : ann a o6 = ideal (d, c, b) o6 : Ideal of A

Currently two algorithms to compute annihilators are implemented. The default is to compute the annihilator of each generator of the module M and to intersect these two by two. Each annihilator is done using a submodule quotient. The other algorithm computes the annihilator in one large computation and is used if Strategy => Quotient is specified.

 i7 : annihilator(M, Strategy => Quotient) 3 2 2 2 3 2 o7 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c) o7 : Ideal of R