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Unformatted text preview: Georgian Mathematical Journal Volume 16 (2009), Number 1, 49–54 ISOMORPHISM CHARACTERIZATION OF DIVISIBLE GROUPS IN MODULAR ABELIAN GROUP RINGS PETER DANCHEV Abstract. Suppose G is an abelian group with a p-subgroup H and R is a commutative unitary ring of prime characteristic p with trivial nil-radical. We give a complete description up to isomorphism of the maximal divisi- ble subgroups of 1 + I ( RG ; H ) and (1 + I ( RG ; H )) /H , respectively, where I ( RG ; H ) denotes the relative augmentation ideal of the group algebra RG with respect to H . This paper terminates a series of works by the author on the topic, first of which are  and . 2000 Mathematics Subject Classification: 16U60, 20K10, 20K21. Key words and phrases: Divisible groups, rank, isomorphism description, torsion units. I. Introduction Throughout the present paper, we assume that G is an abelian group with p- component of torsion elements G p and p-socle G [ p ] and that R is a commutative ring with identity (often called a commutative unitary ring ) of prime charac- teristic p with the ideal of nilpotent elements N ( R ) and with maximal perfect (often termed maximal divisible ) subring dR . For such R and G , where G is written multiplicatively as it is customary when dealing with group rings, RG denotes the group algebra of G over R with the normalized group V ( RG ) of invertible elements in RG (often named as normalized units ) and its subgroup of p-torsion elements S ( RG ). For a subgroup H of G , I ( RG ; H ) denotes the relative augmentation ideal of RG with respect to H . When H ≤ G p , we denote by S ( RG ; H ) the sum 1+ I ( RG ; H ); clearly, S ( RG ; H ) is a subgroup of S ( RG ) and S ( RG ) = S ( RG ; G ) when G = G p . All other notions as well as the notation are standard and follow essentially those from the existing literature with the exception of dG which means the maximal divisible subgroup of G and G * the maximal p-divisible subgroup of G . The isomorphism classification of the maximal divisible subgroup dS ( RG ) of S ( RG ) was started by Mollov–Nachev in  when G is a p-group, and independently by Mollov in  when R is a field. Later, Nachev generalized in  and  these results to arbitrary abelian groups and arbitrary commutative unitary rings of prime characteristic. Concerning the mixed and torsion cases, we classify in  and  dV ( RG ) up to isomorphism assuming that either G is p-mixed and R is a field, or G is a special torsion group and R is a special decomposable or indecomposable ISSN 1072-947X / $8.00 / c Heldermann Verlag www.heldermann.de 50 P. DANCHEV ring, respectively; it is noteworthy that the first result for p-mixed groups was slightly extended by Kuneva  to an indecomposable ring R ....
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