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

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- by Peter Danchev
- Proc. Amer. Math. Soc.
**125**(1997), 2559-2564 - DOI: https://doi.org/10.1090/S0002-9939-97-04052-5
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## Abstract:

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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## Bibliographic Information

**Peter Danchev**- Affiliation: Department of Algebra, Plovdiv University, Plovdiv 4000, Bulgaria
- MR Author ID: 346948
- 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 â€śScientific Researchesâ€ť of the Bulgarian Ministry of Education and Science under contract MM 70/91.
- Communicated by: Ken Goodearl
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 2559-2564 - MSC (1991): Primary 20C07; Secondary 20K10, 20K21
- DOI: https://doi.org/10.1090/S0002-9939-97-04052-5
- MathSciNet review: 1415581