#### 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.

 ```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 ```