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]
-------------------------------
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