Prove that there exist exactly four non-isomorphic rings with 6 elements.

JayDoe

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Prove that there exist exactly four non-isomorphic rings with 6 elements.
 
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What help do you need? Do you know what a "ring" is? Do you know what "isomorphic" means? Since there are only 4 such rings, I would think the easiest thing to do would be to list four rings, show that they are not isomorphic and show that there at no others.

To start with, of course, there must be a "0" and a "1". That leaves 4 other members. Call them "a", "b", "c", and "d". Now think about additive inverses for each. Since additive inverse come in pairs we can have "a and b are inverse to each other" and "c and d are inverse to each other" or we could have "a and b are inverse to each other" while "c is its own inverse" and "d is its own inverse" or we could have "each of a, b, c, and d is its own inverse".
 
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