7.3.22 quotient (plural)

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, N submodules in .Then

• quotient(I,J)= ,
• quotient(M,J)= .

Note:

It can only be used for two-sided ideals (bimodules) in the second argument, otherwise the result may have no meaning.

Example:

//------ a very simple example ------------ ring r=(0,q),(x,y),Dp; def R=nc_algebra(q,0); // this algebra is a quantum plane setring R; option(returnSB); poly f1 = x^3+2*x*y^2+2*x^2*y; poly f2 = y; poly f3 = x^2; poly f4 = x+y; ideal i = f1,f2; ideal I = twostd(i); ideal j = f3,f4; ideal J = twostd(j); quotient(I,J); ==> _[1]=y ==> _[2]=x2 module M = x*freemodule(3), y*freemodule(2); quotient(M, ideal(x,y)); ==> _[1]=gen(1) ==> _[2]=gen(2) ==> _[3]=x*gen(3) kill r,R; //------- a bit more involved example LIB "ncalg.lib"; def Usl2 = makeUsl2(); // this algebra is U(sl_2) setring Usl2; ideal i = e3,f3,h3-4*h; ideal I = std(i); poly C = 4*e*f+h^2-2*h; ideal H = twostd(C-8); option(returnSB); ideal Q = quotient(I,H); // print a compact presentation of Q: print(matrix(Q)); ==> h,f3,ef2-4f,e2f-6e,e3