  | ### 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
See ideal (plural); module (plural).

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