Is R^2 a Field with Component Wise Operations?

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Hello,

Am I missing something or is R^2 a field with the obvious component wise addition and multiplication (a,b)*(c,d)=(ac,bd)?
 
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EDIT: Completely ignore this. Didn't think it through. For an example of why it's false see Office Shredder's reply.

Yes if R is a field, then R^2 is a field (clearly commutative, and (a,b) has inverse (1/a,1/b) ).
 
Last edited:
What's the inverse of (3,0)?
 
Ok, that's what I was missing :)
 
If R, S are non-trivial integral domains, then their direct product R\times S is never an integral domain, because it always has zero-divisors: (a,0).(0,b)=0. In particular, this holds for fields.
 

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