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I tried to study why Krull's intersection theorem won't work in non-Noetherian rings. It was said here that the example written by user2035 works by taking some kind of square-zero ideal. How do one defines a square-zero ideal of a ring? Is it just an ideal $I$ such that $I^2=0$?

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We say that an ideal is a square zero ideal if $I^2=0$.

For example, let $I=(x) \subset k[x]/(x^2)$. Then $I^2=0$.

user26857
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Fredrik Meyer
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