#### 7.2.1.3 ideal operations (plural)

`+`
  addition (concatenation of the generators and simplification)

`*`
  multiplication (with ideal, poly, vector, module; in case of multiplication with ideal or module, the result will be simplified)

`^`
  exponentiation (by a non-negative integer)

ideal_expression `[` intvec_expression `]`
  are polynomial generators of the ideal, index 1 gives the first generator.

Note: For simplification of an ideal, see also simplify.

Example:

    ring r=0,(x,y,z),dp;
    matrix D[3][3];
    D[1,2]=-z;  D[1,3]=y;  D[2,3]=x;
    def R=nc_algebra(1,D); // this algebra is U(so_3)
    setring R;
    ideal I = 0,x,0,1;
    I;
    ==> I[1]=0
    ==> I[2]=x
    ==> I[3]=0
    ==> I[4]=1
    I + 0;    // simplification
    ==> _[1]=1
    I*x;
    ==> _[1]=0
    ==> _[2]=x2
    ==> _[3]=0
    ==> _[4]=x
    ideal J = I,0,x,x-z;
    I * J;   //  multiplication with simplification
    ==> _[1]=1
    vector V = [x,y,z];
    print(I*V);
    ==> 0,x2,0,x,
    ==> 0,xy,0,y,
    ==> 0,xz,0,z
    ideal m = maxideal(1);
    m^2;
    ==> _[1]=x2
    ==> _[2]=xy
    ==> _[3]=xz
    ==> _[4]=y2
    ==> _[5]=yz
    ==> _[6]=z2
    ideal II = I[2..4];
    II;
    ==> II[1]=x
    ==> II[2]=0
    ==> II[3]=1

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