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Let $R$ be a finite ring that satisfies the following conditions:

(1) For any $x\in R$, if $x\ne 0$ then $x^2\ne 0$.

(2) There exists at most one nonzero element $y\in R$ that satisfies $y^2=y$.

How can we show that $R$ has no zero divisors?

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    **Please** do not delete questions and reask them anew: you can *edit* them.2012-12-09
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    This seems to be a homework problem or one out of a book. If that is the case, it is useful that you tell us where it is from, and ideally what you have tried, where you get stuck and other information that will help people help you! It also helps to know if you are thinking about commutative rings and/or rings with unit2012-12-09
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    I don't know where it come from. This problem is given by a friend of me. He don't know either.2012-12-09
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    Ah! The Problem That Came Out of Nowhere. There ought to be a Holmes story with title like that :-)2012-12-09
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    If the ring is commutative the question ammounts to prove the ring is, in fact, a field. Knowing this perhaps can give some insight, yet the answer Mariano wrote below gives a general solution without assuming commutativity.2012-12-09

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