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Let $R$ be a commutative ring such that, for every $a\in R$, the smallest ideal containing $a$ is equal to $Ra$. Does $R$ have identity?

I did this when $R$ has at least one non-zero divisor then it's true.

But I can't guess that it is true in general case.

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No.

Consider $\oplus_{i=1}^\infty F_2$ meaning the subset of $\prod_{i=1}^\infty F_2$ whose elements are only finitely nonzero. And $F_2$ means the field of two elements.

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