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A note on isomorphic commutative group algebras over certain rings. (English) Zbl 1113.20005

Let \(R\) be a commutative ring with identity and let \(G\) be an Abelian group. Denote by \(RG\) the group algebra of \(G\) over \(R\). Suppose \(\text{inv}(R)\) is the set of all rational primes \(p\) such that \(p\) is a unit of \(R\). Following W. Ullery, [Commun. Algebra 14, 767-785 (1986; Zbl 0587.16011)], \(R\) is said to be an ND-ring if whenever \(R\) is written as a finite product of rings, then one of the factors, say \(R_i\), satisfies \(\text{inv}(R_i)=\text{inv}(R)\). The group \(G\) is called \(R\)-favorable if whenever a prime \(p\) is a unit in \(R\), then \(G\) has a trivial \(p\)-component [W. Ullery, Rocky Mt. J. Math. 22, No. 3, 1111-1122 (1992; Zbl 0773.16008)].
The author tries to prove the following result: “Theorem. Suppose \(R\) is a commutative ring with identity of \(\text{char}(R)\neq 0\) and \(G\) is a torsion \(R\)-favorable Abelian group. Then \(RG\cong RH\) as \(R\)-algebras over \(R\) for any group \(H\) imply \(H\) is a torsion \(R\)-favorable Abelian group if and only if either 1) \(\text{inv}(R)=\emptyset\) or 2) \(\text{inv}(R)\neq\emptyset\) and \(R\) is an ND-ring.” (page 69, lines 21-18 from below.)
This paper is unsuccessful. The author gives a definition of an ND-ring \(R\), which is different from the above mentioned definition of Ullery. In the definition of Ullery it is required that every decomposition of \(R\) in a direct product of rings has the given property and in the definition of Danchev that there exists a decomposition of \(R\) in a direct product of rings with this property. Therefore, in the definition of Danchev for an ND-ring there are weaker requirements.
For the proof of the mentioned theorem we can make a series of remarks. The consideration of the cases in the proof of the theorem is confused. The author does not delimit whether he proves the necessity or the sufficiency: for example case 1) is referred to the sufficiency and case 2) has to be referred to the necessity. Such consideration introduces a vagueness in the proof.
Danchev uses the following theorem of Ullery, cited in the paper: “Theorem. ([U]). Suppose \(G\) is an Abelian group and \(R\) is an ND-ring or an indecomposable commutative ring with \(1\). If \(G\) is \(R\)-favorable and \(RH\cong RG\) as \(R\)-algebras for any group \(H\), then \(H\) is \(R\)-favorable as well. Even more, \(H\cong G\) provided \(\text{char}(R)=0\).” (page 68, line 2-1 from below and page 69, line 1-2 from above.)
(i) First of all, we assume, that Danchev does not use his own definition for an ND-ring and uses the definition of Ullery, which is the most probable. Then the sufficiency of Danchev’s theorem follows in a trivial way from the definition of an \(R\)-favorable group and from the cited theorem of Ullery.
The necessity of the theorem is treated entirely confused. Namely, the quotation of the mentioned theorem of Ullery in cases 2.1) and 2.2) is incorrect, since it is referred to the sufficiency.
The necessity should be proved in the following way. Let \(\text{inv}(R)\neq\emptyset\), i.e. it holds case 2) of the paper. If 2.1) \(R\) is an indecomposable ring, then obviously \(R\) is an ND-ring. For the main case 2.2) of the theorem the author writes “we distinguish two possibilities for \(\text{char}(R)\)” (page 69, last line), namely a) and b), but obviously these cases coincide with each other. Furthermore, instead of proving the necessity of the theorem in the general case, the author proves it only in a particular case by an example for a special group \(G\) and a ring \(R\). In this way the necessity remains unproved in the main case 2.2) (see the end of the proof of the theorem).
(ii) If we assume certainly that the author uses his own definition for an ND-ring, then the sufficiency of the theorem could not be proved, since for its proof must be used the result of Ullery, which uses Ullery’s definition of ND-ring, by stronger requirements (in Danchev’s definition there are weaker requirements). Besides, as we mentioned above, the necessity of this theorem is proved only in a particular case by an example for a special group \(G\) and a ring \(R\). – Therefore, in cases (i) and (ii), the theorem of Danchev remains unproved.
In addition, we can remark that, in the formulation of the theorem, the requirement \(\text{inv}(R)\neq\emptyset\) in case 2) is superfluous.
The claims of the author in the abstract that the “strengthened results due to W. Ullery [Commun. Algebra (loc. cit.), Rocky Mt. J. Math. (loc. cit.) and Comment. Math. Univ. Carol. 36, No. 1, 11-14 (1995; Zbl 0828.20005)]” are, to put it mildly, absolutely far-fetched. In Ullery’s mentioned papers are obtained results connected with the isomorphism problem, which is main in the theory of the group algebras and in the article of Danchev’s the main result is only a group theoretic property. In Ullery’s first cited paper is given a sufficient condition for this property to be fulfilled (see the above cited theorem of Ullery) and in Danchev’s article it is a necessary and sufficient condition but through a restrictive condition for the group \(G\) (see the above cited theorem of Danchev).
In conclusion, we may say that the main result of the paper remains unproved.

MSC:

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20K10 Torsion groups, primary groups and generalized primary groups
16U60 Units, groups of units (associative rings and algebras)
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