We know that the only non-zero ring homomorphism from $\mathbb{R}$ to $\mathbb{R}$ is identity. From this some questions came in to my mind as follow:
Question $1$: Can we characterize all fields $F$ which only has the identity function as the ring homomorphism of $F$?
If the answer is "no" in general, can we find a partial answer if we restrict ourselves to subfields of $\mathbb{R}$?
Also we can improve our Question 1 to subrings of $\mathbb{R}$, i.e.
Question $2$: Can we find nice subrings of $\mathbb{R}$ (not necessarily subfields) with the aforesaid property?