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A Group Ring is, as the name suggests, a combination of two simpler algebraic objects, namely groups and rings. A group ring may be thought of as a sort of direct product between a group and a ring. As this might suggest a the group ring RG (R a ring G a group consists of all possible linear combinations of the elements of G over the ring R, and the new hybrid RG is itself a ring (which contains a copy of G in the unit group and a copy of R in the ring itself).

For example if our group G is the group {e,a,b} (The Cyclic group of order 3 with identity e) an element of ZG is 1*e+75*a+33*b. Another example would be 0*e+4*a+2*b, but when we have 0*x we usually choose to omit that term, so the latter may also be written as 4*a+2*b. Also to make the embedding of R clear we traditionally omit the '*e' on the term which is multiplied by the identity, so that if r is an element of r then we may write that r is an element of RG, and we will mean r= r*e+0*g1+0*g2+... and likewise if g is an element of G we may write that g is in RG and we shall by g mean g= 1*g provided that R is a unit ring (traditionally R will be assumed to be a unit ring, or in many cases even a field , but this is not strictly necessarry).

So why group rings? One possibility is that you are interested in representation theory, in which case the group ring is particuarly nice and is often studied as a vector space over the ring R (which is in representation theory taken to be a field). Another possibility is that you are concerned with what types of rings contain a given group as a subgroup of their unit group. How can group rings help then? There is a clear homomorphism from ZG to any ring R containing G which takes G in ZG onto G in R, and hence some subring of R containing G is isomorphic to a quotient ring of ZG via some ideal, and this may well prove useful, especially if additional constraints are known (i.e. the characteristic of the ring for example).

In short group rings get a bad wrap as being unintuive, bulky, and hard to work with, but from the right approach they occur naturally, and can, in the right circumstances, be intuitive and easy to work with, even if they are bulky.

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