# noetherianOperators(Module,Ideal) -- Noetherian operators of a primary component

## Synopsis

• Function: noetherianOperators
• Usage:
noetherianOperators (U, P)
• Inputs:
• U, ,
• P, an ideal, a minimal prime of comodule U
• Outputs:

## Description

Compute a set of Noetherian operators for the $P$-primary component of comodule U.

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : U = image matrix{{x,y,z},{y,z,x}} o2 = image | x y z | | y z x | 2 o2 : R-module, submodule of R i3 : P = first associatedPrimes comodule U o3 = ideal (y - z, x - z) o3 : Ideal of R i4 : noetherianOperators(U, P) o4 = {| -1 |} | 1 | o4 : List

If there are no embedFded primes, running this command for all associated primes is equivalent to running a differential primary decomposition.

 i5 : associatedPrimes comodule U / (P -> {P, noetherianOperators(U,P)}) 2 2 o5 = {{ideal (y - z, x - z), {| -1 |}}, {ideal (x + y + z, y + y*z + z ), {| | 1 | | ------------------------------------------------------------------------ -z |}}} y | o5 : List i6 : netList differentialPrimaryDecomposition U +--------------------------------+--------+ o6 = |ideal (y - z, x - z) |{| -1 |}| | | | 1 | | +--------------------------------+--------+ | 2 2 | | |ideal (x + y + z, y + y*z + z )|{| -z |}| | | | y | | +--------------------------------+--------+