## Commutative group algebras of summable $$p$$-groups.(English)Zbl 1122.20003

Summary: Let $$G$$ be an Abelian group whose $$p$$-component $$G_p$$ is summable and let $$F$$ be a field of characteristic $$p$$. Let $$S(FG)$$ be the normalized unit $$p$$-group of the group algebra $$FG$$. The main results of the present article are that $$G_p$$ is a direct factor of $$S(FG)$$ with totally projective complement, provided $$G_p$$ is of countable length and $$F$$ is perfect; and, in particular, under these circumstances $$S(FG)$$ is summable if and only if $$G_p$$ is summable. Moreover, if $$FG\cong FH$$ as $$F$$-algebras for some group $$H$$ and if $$G_p$$ is summable, then $$H_p$$ is summablale. These achievements improve results due to P. Hill and W. Ullery [Commun. Algebra 25, No. 12, 4029-4038 (1997; Zbl 0901.16012)] and also their generalizations given by P. Danchev [Hokkaido Math. J. 29, No. 2, 255-262 (2000; Zbl 0967.20003); Commun. Algebra 28, No. 5, 2521-2531 (2000; Zbl 0958.20003); ibid. 29, No. 5, 1953-1958 (2001; Zbl 0993.20003); Kyungpook Math. J. 44, No. 1, 21-29 (2004; Zbl 1060.20006)].

### MSC:

 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16U60 Units, groups of units (associative rings and algebras) 16S34 Group rings 20K10 Torsion groups, primary groups and generalized primary groups 20K21 Mixed groups
Full Text:

### References:

 [1] Beers D., Rend. Sem. Mat. Univ. Padova 69 pp 41– (1983) [2] Cutler D. O., Proc. Amer. Math. Soc. 26 pp 43– (1970) [3] Danchev P. V., Compt. Rend. Acad. Bulg. Sci. 46 pp 13– (1993) [4] Danchev P. V., Compt. Rend. Acad. Bulg. Sci. 48 pp 7– (1995) [5] DOI: 10.1090/S0002-9939-97-04052-5 · Zbl 0886.16024 [6] Danchev P. V., Compt. Rend. Acad. Bulg. Sci. 51 pp 13– (1998) [7] Danchev P. V., Math. J. Okayama Univ. 40 pp 77– (1998) [8] Danchev P. V., Hokkaido Math. J. 29 pp 255– (2000) [9] DOI: 10.1080/00927870008826975 · Zbl 0958.20003 [10] Danchev P. V., Compt. Rend. Acad. Bulg. Sci. 53 pp 5– (2000) [11] DOI: 10.1081/AGB-100002160 · Zbl 0993.20003 [12] Danchev P. V., Compt. Rend. Acad. Bulg. Sci. 55 pp 5– (2002) [13] Danchev P. V., Math. J. Okayama Univ. 45 pp 1– (2003) [14] Danchev P. V., Kyungpook Math. J. 44 pp 21– (2004) [15] Fuchs L., Infinite Abelian Groups. I (1974) · Zbl 0274.20067 [16] Fuchs L., Infinite Abelian Groups II (1977) [17] Hill P. D., Proc. Amer. Math. Soc. 23 pp 428– (1969) [18] DOI: 10.1216/RMJ-1971-1-2-345 · Zbl 0229.20054 [19] Hill , P. D. ( 1977 ).Criteria for Freeness in Groups and Valuated Vector Spaces. Lecture Notes in Math. Berlin-New York : Springer Verlag, 616 , pp. 140 – 157 . · Zbl 0372.20041 [20] DOI: 10.1090/S0002-9939-1990-1039530-4 [21] DOI: 10.1080/00927879708826107 · Zbl 0901.16012 [22] Kurosh A. G., The Theory of Groups (1967) · Zbl 0189.30801 [23] DOI: 10.1090/S0002-9947-1969-0233903-9 [24] May W. L., Illinois J. Math. 15 pp 525– (1971) [25] May W. L., Proc. Amer. Math. Soc. 76 pp 31– (1979) [26] DOI: 10.1090/S0002-9939-1988-0962805-2 [27] May W. L., Contemp. Math. 93 pp 303– (1989) [28] DOI: 10.4153/CJM-1969-132-9 · Zbl 0208.03502 [29] Mollov T. Zh, Pliska 8 pp 54– (1986) [30] DOI: 10.1007/BF01135839 · Zbl 0173.02401 [31] Rangaswamy K. M., Bull. Soc. Math. France 92 pp 259– (1964) [32] DOI: 10.1016/0021-8693(79)90330-2 · Zbl 0401.20049 [33] DOI: 10.1080/00927879208824365 · Zbl 0749.16016 [34] Wallace K. D., Pacific J. Math. 43 pp 799– (1972) · Zbl 0246.20045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.