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hsong9
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Homework Statement
Let G be a finite group and let p >= 3 be a prime such that p | |G|.
Prove that the group ring ZpG is not a domain.
Hint: Think about the value of (g − 1)p in ZpG where g in G and where
1 = e in G is the identity element of G.
The Attempt at a Solution
G is a finite ring and p is a prime number such that p divides the order of G for k times.
since p is a prime less than the order of G , there exists an element a in G such that (ap) k = n where n is the order of G.
there exists an element (a m ) p such that ( a m )p* (am)q = 0
so, (a m )p and (am)q are the zero divisors in Zp G.
∴ Zp(G) is not an integral domain.