Exercise III.3.11 (Aluffi): Let $R$ be a ring containing $\mathbb{C}$ as a subring. Prove that there are no (non-zero) ring homomorphisms $R \rightarrow \mathbb{R}$.
I thought this was a cool exercise that generalizes the fact that there are no (non-zero) ring homomorphisms from $\mathbb{C}$ to $\mathbb{R}$. I wanted to hear any commments/advice for improvement on the proof (as well as if the logic is correct).
Lemma: There are no (non-trivial) ring homomorphisms from $\mathbb{C}$ to $\mathbb{R}$.
Proof: For the sake of contradiction, let $\varphi: \mathbb{C} \rightarrow \mathbb{R}$. Then let $\varphi(i) = r$ for some $r \in R$. It then follows that since $\varphi(0_\mathbb{C}) = 0_\mathbb{R}$ and $\varphi(1_\mathbb{C}) = 1_\mathbb{R}$ that \begin{equation*} 0_\mathbb{R} = \varphi(i^2 + 1_\mathbb{C}) = \varphi(i)^2 + \varphi(1_\mathbb{C}) = r^2 + 1 \end{equation*} which is impossible as $r^2 \geq 0$ when $r \in \mathbb{R}$. Thus by contradiction the only ring homomorphism from $\mathbb{C}$ to $\mathbb{R}$ is the trivial ring homomorphism.
(Thus the proof of the Exercise is immediate.)
Proof (of Exercise): For the sake of contradiction, assume that there is a non-zero ring homomorphism, $\varphi: R \rightarrow \mathbb{R}$, where $R$ is a ring with containing $\mathbb{C}$ as a subring. Since $\mathbb{C}$ is a subring, it follows that the inclusion function $\iota: \mathbb{C} \hookrightarrow R$ is a ring homomorphism. Thus, it follows that $(\varphi \circ \iota): \mathbb{C} \rightarrow \mathbb{R}$ is a non-zero ring homomorphism. However, this contradicts the fact that there are no non-zero ring homomorphisms from $\mathbb{C}$ to $\mathbb{R}$, and the exercise follows.
While I found this small exercise neat and fun to do, I feel as if the proof I provided could have been slightly more efficient. Any advice?
Also, some facts I feel do not have to be explicitly stated, such as the composition of two non-zero ring homomorphism is itself non-zero, or that $R$ is not the zero ring as it contains $\mathbb{C}$. I notice as I slowly trod towards "higher" levels of math, some details which seem obvious to me, might not be obvious to all readers. How does one decide what to explicitly state within their proofs in higher level math? Sorry if this line of questions warrant their own thread, I just want to be able to articulate arguments in the best way possible moving forward (always looking for refinements to my proof writing technique).