Singular Manual: kbase (plural)

  | ### 7.3.10 kbase (plural) `Syntax:` `kbase (` ideal_expression `)` `kbase (` module_expression `)` `kbase (` ideal_expression`,` int_expression`)` `kbase (` module_expression`,` int_expression`)` `Type:` the same as the input type of the first argument `Purpose:` with one argument: computes the vector space basis of the factor-module that equals ring (resp. free module) modulo the ideal (resp. submodule), generated by the initial terms of the given generators. If the factor-module is not of finite dimension, -1 is returned. If the generators form a Groebner basis, this is the same as the vector space basis of the factor-module. when called with two arguments: computes the part of a vector space basis of the respective quotient with degree (of monomials) equal to the second argument. Here, the quotient does not need to be finite dimensional. `Note:` in the non-commutative case, a ring modulo an ideal has a ring stucture if and only if the ideal is two-sided. `kbase` respects module-grading given by the `isHomog` attribute of input modules. `Example:`
      ring r=0,(x,y,z),dp;
      matrix d[3][3];
      d[1,2]=-z;  d[1,3]=2x;  d[2,3]=-2y;
      def R=nc_algebra(1,d); // this algebra is U(sl_2)
      setring R;
      ideal i=x2,y2,z2-1;
      i=std(i);
      print(matrix(i));  // print a compact presentation of i
      ==> z2-1,yz-y,xz+x,y2,2xy-z-1,x2
      kbase(i);
      ==> _[1]=z
      ==> _[2]=y
      ==> _[3]=x
      ==> _[4]=1
      vdim(i);
      ==> 4
      ideal j=x,z-1;
      j=std(j);
      kbase(j,3);
      ==> _[1]=y3
See ideal (plural); module (plural); vdim (plural).

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