Characterization of Integral Domains

Question

We can show that in an Integral Domain I if $x^2=1$ then $x=pm 1$.
Is the converse true?
i.e if for any x in I with $x^2=1 implies x=pm1$ then I is an Integral Domain.

user10354138 · Answer

No.  Take $mathbb{Z}/4$, for example, where it is well-known all squares are 0 or 1 (depending on how they reduce in $mathbb{Z}/2$).  So $x^2=1$ iff $x=pm 1$ but this isn't an integral domain.

Characterization of Integral Domains

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