Let $R$ be a commutative finite ring in which $ab = 0$ implies either $a = 0$ or $b = 0$ for any $a,b \in R$. Then, $R$ is a field.
I do not understand how I should act. I tried different ways, but I was not able to prove this assertion.
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1You need additional conditions on $R$, a word was probably left out. – André Nicolas Dec 26 '14 at 23:06
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1Can someone explain to me why this is downvoted? – k.stm Dec 26 '14 at 23:06
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2@k.stm: My best guess is that someone thought this looked like the OP was outsourcing his mathematical work. – Dec 26 '14 at 23:08
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@someoneyouknow, I made a mistake. Z is not a finite ring. – Max Dec 26 '14 at 23:12
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For further variations see answers here. This is one of the most frequently asked questions. – Bill Dubuque Dec 26 '14 at 23:18
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What you are trying to prove is that a commutative ring with no zero divisors is a field. This is false unless $R$ is finite. If $R$ is finite, the only thing you have to show is that every non-zero element of $R$ has an inverse. Then the solution is for example here: How to show that a finite commutative ring without zero divisors is a field?
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