Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
unit groups; isomorphism; Ulm-Kaplansky invariants; commutative twisted group rings
unit groups; isomorphism; Ulm-Kaplansky invariants; commutative twisted group rings
Summary:
Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_tG$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S(R_tG)$ are the $p$-components of $G$ and of the unit group $U(R_tG)$ of $R_tG$, respectively. Let $R^*$ be the multiplicative group of $R$ and let $f_\alpha (S)$ be the $ \alpha $-th Ulm-Kaplansky invariant of $S(R_tG)$ where $\alpha $ is any ordinal. In the paper the invariants $f_n(S)$, $ n\in \mathbb{N}\cup \lbrace 0\rbrace $, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said to be multiplicatively $p$-perfect if $(R^*)^p = R^*$. For these rings the invariants $f_\alpha (S)$ are calculated for any ordinal $\alpha $ and a description, up to an isomorphism, of the maximal divisible subgroup of $S(R_tG)$ is given.
References:
[1] S. D. Berman: Group algebras of countable abelian $p$-groups. Publ. Math. 14 (1967), 365–405. (Russian) MR 0225897 | Zbl 0178.02702
[2] A. A. Bovdi and S. V. Mihovski: Idempotents in crossed products. Bull. Institut Math. Acad. Bulgaria 13 (1972), 247–263. (Russian) MR 0439917
[3] A. A. Bovdi and Z. F. Pataj: On the construction of the centre of a multiplicative group of the group ring of the $p$-groups over a ring of characteristic $p$. Proc. Bielorus. Acad. Sci. 1 (1978), 5–11. (Russian)
[4] L. Fuchs: Infinite Abelian Groups, Vol. 1. Publishing House World, Moscow, 1977. (Russian)
[5] T. Zh. Mollov: On the unit groups of the modular group algebras of primary abelian groups of an arbitrary cardinality I. Publ. Math. Debrecen 18 (1971), 9–21. (Russian) MR 0311779
[6] T. Zh. Mollov: Ulm invariants of the Sylow $p$-subgroups of the group algebras of the abelian groups over a field of characteristic $p$. Sixth Congress of the Bulgarian Mathematicians, Varna, 1977, Reports Abstracts, Section A2, p. 2. (Bulgarian) MR 0633857
[7] T. Zh. Mollov: Ulm invariants of the Sylow $p$-subgroups of the group algebras of the abelian groups over a field of characteristic $p$. Pliska 2 (1981), 77–82. (Russian) MR 0633857
[8] T. Zh. Mollov and N. A. Nachev: Some set theoretic properties of the radical of Baer of commutative rings of prime characteristic. Plovdiv University “P. Hilendarsky”, Scientific works 15, book 1 (1977). (Russian)
[9] T. Zh. Mollov and N. A. Nachev: On the semisimple twisted group algebras of primary cyclic groups. Houston J. Math. 25 (2000), 55–66. MR 1814727
[10] N. A. Nachev: Invariants of the Sylow $p$-subgroup of the unit group of commutative group ring of characteristic $p$. Comm. Algebra 23 (1995), 2469–2489. DOI 10.1080/00927879508825355 | MR 1330795
[11] N. A. Nachev and T. Zh. Mollov: Ulm-Kaplansky invariants of the groups of normalized units of the modular group ring of a primary abelian group. Serdica 6 (1980), 258–263. (Russian) MR 0608404
[12] N. A. Nachev and T. Zh. Mollov: Sylow $p$-subgroups of commutative twisted group algebras of the finite abelian $p$-groups. Serdica 14 (1988), 161–178. (Russian) MR 0949198
[13] N. A. Nachev and T. Zh. Mollov: Semisimple twisted group algebras of cyclic $p$-groups of odd order. Publ. Math. Debrecen 37 (1990), 55–64. (Russian) MR 1063657
[14] D. S. Passman: The Algebraic Structure of Group Rings. Wiley-Interscience, New York, 1977. MR 0470211 | Zbl 0368.16003
[15] D. S. Passman: Infinite Crossed Products. Acad. Press, San Diego, 1989. MR 0979094 | Zbl 0662.16001
[16] W. Ullery: An isomorphism theorem for commutative modular group algebras. Proc. Amer. Math. Soc. 110 (1990), 287–292. DOI 10.1090/S0002-9939-1990-1031452-8 | MR 1031452 | Zbl 0712.20036
Search
Partner of
