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Problems Pertaining to Group Rings

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a. Let G = {e, a} where a^2 = e. Clearly G is isomorphic to Z_2. We claim that the quotient ring
S = Z[X]/(2, X^2 - 1) is isomorphic to R[G]. To see this, consider the map phi: R[G] -> S which satisfies phi(0) = 0, phi(e) = 1, phi(a) = X and phi(a + e) = 1 + X. Clearly phi is a set isomorphism, so we merely need to show that it is a ring homomorphism, i.e.

(i) phi(s_1 + s_2) = phi(s_1) + phi(s_2)
(ii) phi(s_1 s_2) = phi(s_1) phi(s_2)

for all elements s_1 and s_2 of S.

Consider Equation (i) first. There are 16 cases we need to check. Equation (i) clearly holds when either s_1 or s_2 are equal to zero since phi(0) = 0. There are 9 cases left to consider. Since g + g = 2g = 0 for all g in G, we see that phi(g + g) = phi(0) = 0 = 2 phi(g) = phi(g) + phi(g). This eliminates 3 of the 9 possibilities, so we have 6 cases left. We also note that both R[G] and S are commutative with respect to addition, so this narrows down the number of cases to just 3. We check these cases by hand. We have

phi(a + e) = 1 + X ...

Solution Summary

We solve several problems pertaining to group rings, i.e. rings formed by multiplying group elements by elements of a fixed ring.