Rings and equations

Question

Let $R$ be a commutative (non-zero) ring with identity, what are the solutions of $x^2-1=0$? Obviously, $x=\pm 1$ are solutions and if $R$ is an integral domain there aren't other solutions since $x^2-1=(x-1)(x+1)$. Anyway lately I found a text which assert without proof that in a commutative ring with identity there are only these two solutions: is it just a mistake or not?

Answer 1

It’s a mistake, the factorization $X^2 - 1 = (X-1)(X+1)$ characterizes the rings with more solutions: They have to contain a pair of zero divisors which differ by $2$. For example in a ring of characteristic $2$ with nilpotent elements of order $2$ like $\mathbf F_2 [T]/(T^2)$ the solutions to $X^2 - 1$ are not only $\pm 1$, but also $T+1$ and $T-1$, but there are plenty other rings which refute the claim.