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Commutative group algebras of $\sigma$-summable abelian groups

Author: Peter Danchev
Journal: Proc. Amer. Math. Soc. 125 (1997), 2559-2564
MSC (1991): Primary 20C07; Secondary 20K10, 20K21
MathSciNet review: 1415581
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In this note we study the commutative modular and semisimple group rings of $\sigma$-summable abelian $p$-groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that $S(RG)$ is $\sigma$-summable if and only if $G_p$ is $\sigma$-summable, provided $G$ is an abelian group and $R$ is a commutative ring with 1 of prime characteristic $p$, having a trivial nilradical. If $G_p$ is a $\sigma$-summable $p$-group and the group algebras $RG$ and $RH$ over a field $R$ of characteristic $p$ are $R$-isomorphic, then $H_p$ is a $\sigma$-summable $p$-group, too. In particular $G_p\cong H_p$ provided $G_p$ is totally projective of a countable length.

Moreover, when $K$ is a first kind field with respect to $p$ and $G$ is $p$-torsion, $S(KG)$ is $\sigma$-summable if and only if $G$ is a direct sum of cyclic groups.

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Additional Information

Peter Danchev
Affiliation: Department of Algebra, Plovdiv University, Plovdiv 4000, Bulgaria
MR Author ID: 346948

Keywords: Commutative modular and semisimple group algebras, $\sigma$-summable groups, normalized units, isomorphism, totally projective groups
Received by editor(s): March 3, 1995
Received by editor(s) in revised form: April 12, 1996
Additional Notes: This research was supported by the National Foundation โ€œScientific Researchesโ€ of the Bulgarian Ministry of Education and Science under contract MM 70/91.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society