Let $T_{[0,1]}$ denote the ring of all functions $f: [0,1] \rightarrow \mathbb{R}$ and $T_{[0,2]}$ denote the ring of all functions $f: [0,2] \rightarrow \mathbb{R}$. Are they isomorphic?
I assumed they are, since I cannot think of a reason they wouldn't be. I denoted a function f $f:T_{[0,1]} \rightarrow T_{[0,2]}$ to be a function where for $t_{[0,1]}(x)$, $f(t_{[0,1]}(x)) = t_{[0,2]}(2x)$. I am not sure how to proceed from here.
Would the function f(x) = 2x, where x is in $T_{[0,1]}$ be a better start?