1. ## automorphism mappings

Let R be ring, I ideal of R
Let f is automorphism of R.
let say: (f(x)+4 )f(y)=0, $\forall$ x,y $\in$ I
if i say ${f}^{-1 }$ (f(x)+4 ) y = 0, $\forall$ x,y $\in$

whether the statement was valid?
if yes, any reason to explain that? (i know f is invertible cause f is 1-1, but whether there is a more specific explanation?)

2. i don't see anything wrong with that.

since $f^{-1}$ is also an automorphism, you can even say that (x + $f^{-1}$(4))y = 0 for all x,y in I.