Elements in $\mathbb{Z}[S_3]$ are linear combinations of permutation but what does it even mean to have $2\cdot(132)-id+8\cdot(13)$ ?
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Nope actually that would just be $(132)(132) = (123)$ and that is composition within the group... – John Cataldo Feb 27 '18 at 19:02
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The main application of group rings is in representation theory of groups. They give a way of translating group theoretic problems into linear algebra problems, since the theory of what you can do with matrices is very advanced (not only matrices over fields, but also over PIDs.)
See also Do group rings appear outside of representation theory? and Doubt: "A group representation is exactly like a module over the group ring".
The elements of a group algebra do not really have "meaning" as far as I know. You may have some exotic idea in mind (perhaps a physical interpretation) but I am not sure what you want in that case.
Of course, they can be interpreted as functions from $G\to \mathbb Z$, but I don't know what a function "means."
rschwieb
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And here that function's image of an element in G is the coefficient in R that is besides the element in the group? namely $f((id))=-1, f((132))=2$ and $f((13))=8$ ? – John Cataldo Feb 27 '18 at 19:24
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