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I was struggling for days with this nice problem:

Let $A$ be a finite commutative ring such that every element of $A$ can be written as product of two elements of $A$. Show that $A$ has a multiplicative unit element.

I need a hint for this problem, thank you very much.

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    @Theo Buehler : What do you mean ? , I have checked Matt E's and André Nicolas's answers and I don't think that they solved my problem2011-08-09
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    @mathfan, you've asked four questions here previously, and not accepted the answers on any of them. If you didn't like those answers, why do you ask here again? Or if you're not familiar with the concept of "accepting answers", may I suggest you have a look at the faq?2011-08-09
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    I'm sorry I made a slip of basic logic.2011-08-09
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    @Theo: Welcome to the club! :)2011-08-09
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    It suffices that $A$ be finitely generated. (See Jonas’s answer.)2011-08-10

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