Use R ::= QQ[x,y];
I := Ideal(x+y^2, x^2-xy);
GenRepr(x^3-x^2y-y^3-xy, I);
[-y, x]
-------------------------------
-y I.Gens[1] + x I.Gens[2];
x^3 - x^2y - y^3 - xy
-------------------------------
GenRepr(x+y, I);
[ ]
-------------------------------
x+y IsIn I; -- the empty list was returned above since x+y is not in I
False
-------------------------------
V1:= Vector(x, y, y^2); V2:= Vector(x-y, 0, x^2);
X := x^2*V1 - y^2*V2;
M := Module(V1, V2);
GenRepr(X, M);
[x^2, -y^2]
-------------------------------
|