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Isomorphism of group algebras. (English) Zbl 0587.16011

Let R be a commutative ring with 1 and let inv(R) be the set of all rational primes p such that p is a unit of R. Given an abelian group G, let \(G_ R\) denote the direct sum of the p-components of G with \(p\in inv(R)\). Following the author, the ring R is said to satisfy the Isomorphism Theorem if RG\(\cong RH\) implies \(G/G_ R\cong H/H_ R\). Again, following the author, R is said to be an ND-ring (for ”nicely decomposing”) if whenever R is written as a finite product of rings, then one of the factors, say \(R_ i\), satisfies \(inv(R_ i)=inv(R).\)
May’s conjecture asks for a proof or a counterexample to the assertion that all rings satisfy the Isomorphism Theorem [see G. Karpilovsky, Commutative group algebras (1983; Zbl 0508.16010), p. 201, Problem 17].
In this paper it is proved that a ring which satisfies the Isomorphism Theorem is necessarily an ND-ring. For a wide variety of rings R, it is easily checked whether or not R is an ND-ring. This enables the author to construct counterexamples to May’s conjecture. The following is the main result of the paper. Theorem. Suppose inv(R) contains all prime numbers or excludes at least two prime numbers. Then R satisfies the Isomorphism Theorem if and only if R is an ND-ring.
Reviewer: G.Karpilovsky

MSC:

16S34 Group rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)

Citations:

Zbl 0508.16010
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References:

[1] Karpilovsky G., Commutative Group Algebras (1983) · Zbl 0508.16010
[2] DOI: 10.1090/S0002-9947-1969-0233903-9 · doi:10.1090/S0002-9947-1969-0233903-9
[3] DOI: 10.1016/0021-8693(76)90049-1 · Zbl 0328.16012 · doi:10.1016/0021-8693(76)90049-1
[4] DOI: 10.1016/0021-8693(76)90083-1 · Zbl 0329.20002 · doi:10.1016/0021-8693(76)90083-1
[5] Pierce R.S., Memoirs Amer. Math. Soc 70 (1967)
[6] Ullery W., The isomorphism problem for commutative group algebras (1983) · Zbl 0596.16012
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