In Lang's Algebra, he gives a definition that
Elements $x_1, \cdots, x_n\in B$ are called algebraically independent over $A$[a subring of $B$] if the evaluation map $$f\mapsto f(x)$$is injective. Equivalently, we could say that if $f\in A[X]$ is a polynomial and $f(x)=0$, then $f=0$.
I am confused about the "injective" here. Two possible interpretations in my mind:
- Fix an $f$, for $x\neq y,f(x)\neq f(y).$
- Fix an $x$, for $f_1\neq f_2,f_1(x)\neq f_2(x).$
I was wondering which one is correct and why. Could you give me some helpful examples?
Besides, why are these two definitions equivalent?
Thanks in advance.