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I can tell there are very different requirements between group definitions and ring definitions, but my teacher said this statement"If G is a group, that implies Rg is a ring(called group ring of G over R/C/Q/Z)" I know that examples of rings include R,C,Q, and Z, but how does being a group imply being a ring since I'd imagine not all groups are closed under both multiplication and addition. Therefore, my real confusion is are all groups rings or all are rings groups? If I were to hypothesize, I'd say the latter is correct, since rings would always be an abelian group under addition. Is there an example of a group that isn't a ring?

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The group ring $RG$ is a ring that is constructed out of a group $G$. It is different from $G$. So the statement is not that $G$ is a ring, the statement is that $G$ can be used to define a different object $RG$ and this new object is a ring.

Wikipedia has a decent sized article on group rings if you'd like to read about how they are constructed.

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  • $\begingroup$ I looked over wikipedia, so is it correct to say then that a ring over a group is a ring constructed using inputs of group elements? $\endgroup$ – cakey Dec 3 '13 at 14:54
  • $\begingroup$ @cakey And group rings are not far from semigroup rings, which are defined essentially the same way. The prototypical semigroup ring is the ring of polynomials over $R$: I find that helps with the intuition at the beginning. $\endgroup$ – rschwieb Dec 3 '13 at 14:54
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A group ring is just the ring of polynomials but using group elements instead of variables. Additiin is just additiin of polynomials, and multiplication is multiplication of polynomials, except every time two group elements are multiplied you use the groups binary operation.

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