  | ### 7.3.30 vdim (plural) `Syntax:` `vdim (` ideal_expression `)` `vdim (` module_expression `)` `Type:` int `Purpose:` computes the vector space dimension of the factor-module that equals ring (resp. free module) modulo the ideal (resp. submodule), generated by the leading terms of the given generators. If the factor-module is not of finite dimension, -1 is returned. If the generators form a left Groebner basis, this is the same as the vector space dimension of the left factor module. `Note:` In the non-commutative case, a ring modulo an ideal has a ring structure if and only if the ideal is two-sided. `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 RS=nc_algebra(1,d); //U(sl_2)
      setring RS;
      option(redSB); option(redTail);
      ideal I=x3,y3,z3-z;
      I=std(I);
      I;
      ==> I[1]=z3-z
      ==> I[2]=y3
      ==> I[3]=x3
      ==> I[4]=y2z2-y2z
      ==> I[5]=x2z2+x2z
      ==> I[6]=x2y2z-2xyz2-2xyz+2z2+2z
      vdim(I);
      ==> 21
See also ideal (plural); kbase (plural); std (plural).

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