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Trivial units in commutative group algebras. (English) Zbl 1163.16019

Let \(RG\) be the group algebra of an Abelian group \(G\) over a commutative ring \(R\) with identity and let \(V(RG)\) be the group of the normalized units in \(RG\). Suppose \(\text{inv}(R)\) is the set of all rational primes \(p\) such that \(p\) is a unit in \(R\) and \(\text{supp}(G)=\{p\mid G_p\neq 1\}\).
The author tries to give necessary and sufficient conditions for the equality \(V(RG)=G\) when one of the following conditions holds: (i) \(\text{supp}(G)\cap\text{inv}(R)\) is not empty; (ii) \(RG\) is a modular group algebra; (iii) the ring \(R\) has finite characteristic greater than 1.
However, these results remain in general unproved because of the following reasons. For the proof of the results the author uses (page 55, lines 1-2 from above) the proof of the main result of his paper [Math. Commun. 10, No. 2, 143-147 (2005; Zbl 1097.16007)]. However, the last result remains unproved since in its proof the case \((F(\eta_q):F)\neq q-1\) is not considered (\(\eta_q\) is a primitive \(q\)-th root of unity) (only the case “\(q-1=(F(\eta_q):F)\)” is considered in p. 142, line 7 from below). For example, the case \((F(\eta_q):F)\neq q-1\) arises when \(F=\mathbb{Z}_{19}\) and \(q=5\). Namely, it is not hard to see that \((\mathbb{Z}_{19}(\eta_5):\mathbb{Z}_{19})=2\neq 4=q-1\).
We note additionally that for an idempotent \(e\) and for an element \(r\in R\) the author incorrectly writes \(e\neq\{0,1\}\) (page 53, line 9 from above), \((0,1)\neq e^2\) (page 54, line 4 from above) and \(r=(0,1)\) (page 55, line 11 from above). Obviously, these notations are senseless, since any element is always different from any set which consists of two elements and never is equal to such a set.

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20K21 Mixed groups

Citations:

Zbl 1097.16007
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