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3.21 quotient

Syntax:
quotient ( ideal_expression, ideal_expression )
quotient ( module_expression, module_expression )
Type:
ideal
Syntax:
quotient ( module_expression, ideal_expression )
Type:
module
Purpose:
computes the ideal quotient, resp. module quotient. Let R be the basering, I,J ideals and M a module in ${\tt R}^n$.Then
  • quotient(I,J)= $\{a \in R \mid aJ \subset I\}$,
  • quotient(M,J)= $\{b \in R^n \mid bJ \subset M\}$.
Note:
Should be used for two-sided ideals (bimodules) only, otherwise the result may have no meaning.

Example:
 
// from [BGV], Ex.3.17, p.150
ring r=0,(x,y),Dp;
ncalgebra(2,0);
option(returnSB);
poly f1  = x^3+2*x*y^2+2*x^2*y;
poly f2  = y;
poly f1' = x^2;
poly f2' = x+y;
ideal i = f1,f2;
ideal I = twostd(i);
ideal j = f1',f2';
ideal J = twostd(j);
quotient(I,J);
==> _[1]=y
==> _[2]=x2
kill r;
// a bit more complicated example
LIB "ncalg.lib";
def Usl2 =sl2();
setring Usl2;
ideal i = e3,f3,h3-4*h;
ideal I = std(i);
poly  C = 4*e*f+h^2-2*h;
ideal H = C-8;
option(returnSB);
ideal Q = quotient(I,H);
Q;
==> Q[1]=h
==> Q[2]=f3
==> Q[3]=ef2-4f
==> Q[4]=e2f-6e
==> Q[5]=e3
See ideal; module.


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