# Confusion with this statement about rings vs. groups

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?

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.
• @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. – rschwieb Dec 3 '13 at 14:54