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I'm asking this because I could only think of infinite rings where this is true. This must include rings.

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Are you saying you want a ring that does not have any of those elements, or a ring that might have some of them and also has others that do not match those descriptions? A ring is closed under both of its operations, and while it is not guaranteed to have a multiplicative identity, it must have an additive identity... –  abiessu Dec 10 '13 at 20:43

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There's a reason why you can't find one! If by "ring" you mean a ring with unity, then every non-zero element of a finite ring $R$ is either a unit or a zero divisor. See this link for a proof.

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Thank you Dan I didn't realize it was already a proof. –  MathematicalAnomaly Dec 10 '13 at 20:55
    
I would be interested in seeing a proof where there isn't a unity element in the ring... –  abiessu Dec 10 '13 at 20:58
    
@ abiessu: then how do you define a unit or "unitary element"? –  Robert Lewis Dec 10 '13 at 21:08
    
@abiessu Such a ring is necessarily made up entirely of zero divisors, like $2\Bbb Z/8\Bbb Z$. If you can use chat I could sketch the proof there. –  rschwieb Dec 10 '13 at 23:52
    
I thought about it and found a partial proof myself... Thank you for the example :-) –  abiessu Dec 11 '13 at 5:01

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