Remark

Of course different sets of generators V may lead to the same ideal. Here is an example showing that it may be hard to detect whether two subsets of R generate the same ideal.

Take R = Q[X,Y], and

V = {X²Y-1, XY²-1},   W = {X-Y, X³-1}.

We claim (V) = (W).

To see this, we write

X-Y	=	Y(X2Y-1) - X(XY2-1)
X3-1	=	(X2Y+1)(X2Y-1)-X3(XY2-1),

from which we derive that V is contained in (W), which implies that (V) is contained in (W).

Conversely,

X2Y-1	=	-X2 (X-Y) + (X3-1)
XY2-1	=	(-XY-X2)(X-Y)+ (X3-1) ,

whence the equality (V) = (W).

We emphasize that we have not made clear how these expressions are found. This is too difficult for the present notes.