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This isn't there in book. I am just curious.

Let $R$ be a ring, then an ideal $I\subset R$ is called a primary ideal if $ab \in I$ implies either $a\in I$ or $b^n \in I$ for some $n\in \mathbb{N}$.

Now I wanted to modify the definition.

Let $R$ be a ring, then call an ideal $I\subset R$ a secondary ideal if $ab\in I$ implies either $a^m\in I$ for some $m\in \mathbb{N}$ or $b^n \in I$ for some $n\in \mathbb{N}$.

I am wondering if these two definitions of primary ideal and secondary ideal equivalent? Neither I was able to prove it nor I am able to find a counterexample

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    Putting "primary ideal definition" in the search bar gets you three duplicates in the top 10 or so posts. The proposed definition is strictly weaker, and an example is given to show this. – rschwieb Dec 19 '22 at 16:34

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