I was reading Springer's Linear Algebraic Groups and I came across this problem 1.3.9 on page 6.
Let $k = \mathbb{C}$, $\;F = \mathbb{R}$, $\;k[X] = k[T,U]/(T^2 + U^2 -1)$, and let $a,b$ be the images of $T,U$ in $k[X]$.
Show that $F[a,b]$ and $F[ia,ib]$ define two different $F$ structures on $X$.
I can show that they are both $F$-structures. However, I do not see how they are different or what it even means for $F$-structures to be different.
Here is the intuition. $F[a, b]$ corresponds geometrically to the circle $x^2+y^2-1=0$ over $\mathbb{R}$. But $F[ia, ib]$ corresponds to $-x^2-y^2-1=0$, i.e. $x^2+y^2=-1$ over $\mathbb{R}$. This does not have any solutions over $\mathbb{R}$, but the circle has many solutions (a circle worth of solutions).
In short, $F[a, b]$ and $F[ia, ib]$ defines two different $F$-structures on $X$ because the $F$-points of the first one is non-empty (namely, the circle in $\mathbb{R}$), while the second one has no $F$-point. I guess the key here is that two same $F$-structures should certainly have same number of $F$-points. Okay, hopefully someone will give a more complete answer!