# In a commutative ring $R$ if, for every $a\in R$, the smallest ideal containing $a$ is equal to $Ra$ then $R$ has identity?

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.

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.