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I'm making exercises to prepare for my ring theory exam.
Let $R$ be an infinite commmutative ring which contains a zero divisor. Show that there exists infinite many zero divisors.
Let $a\in R$ a zero divisor. Then $a⋅b=0$ for some element $b≠0$ in $R$. If $a$ or $b$ has infinite order, then I can find infinite zero divisors. But otherwise, I don't see why this should be true. I think I need to do something with the fact that $R$ is commutative. A hint or a detailed solution are both appreciated.