Let us consider the power series in two variables $F(x,y)$ over a field $F$ such that $F(x,y)$ converges for $x,y \in U \subset F$.
Then what does $(x,y)^2$ in the representation $$F(x,y) \equiv x+y \mod (x,y)^2$$ mean here?
Does it mean " we are taking modulo by the $\text{term}$ generated by $x,y$ ?
i.e., Is $(x,y)^2=g(x,y)$ a function of $x,y$ ?
or is it that $x^2, y^2, x^2y^2, x^2y^4, x^4y^2, x^4y^4, \cdots \in (x,y)^2$ ?
For example, if we take $F(x,y)=x+y+x^2+y^2+x^2y^2+x^4y^2+x^2y^4+\cdots$, then $$F(x,y) \equiv x+y \mod (x,y)^2.$$
Am I right?