Find the cardinality of a unital ring with an interesting property

Question

Let $(A,+,\cdot)$ be a unital ring with $|A| \ge 4$. If $\forall x,y \in A \setminus \{0,1\}, x\ne y$, we have that either $x^2=y$ or $y^2=x$, find $|A|$.
I didn't make much progress, but I have a hunch that $|A|=4$ since we are given that lower bound. I thought that maybe we should assume that the ring has at least $5$ elements, so we could pick three pairwise distinct elements $x, y, z$, but I didn't seem to reach any contradiction.
I also thought that maybe we may consider the function $f: A \to A, f(x)=x^2$, but I don't know what to do next.
Note: we aren't given that $A$ is finite.

Answer 1

A partial answer.

Consider $2 \in A$ and $3 \in A$. There are three possibilities:

• $2=3$ — in this case $1=0$ and $A$ is the zero ring, so $|A|=1$
• $2^2=3$ — thus, again, $1=0$
• $3^2=2$ — that is, $7=0$, so $A$ is an algebra over $\mathbb F_7$ (the field with 7 elements). Consider $x=2 \in A$ and $y=5 \in A$: $2^2=4$ and $5^2=(-2)^2=4$ (remember that we work modulo $7$ here). In particular, $x\neq y$, yet $x^2 \neq y$ and $x \neq y^2$.
• $2=0$
• $3=0$

Thus, we are forced to assume either $2=0$ or $3=0$.