Geodesic
On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive
Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1365-1404 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the present paper certain methods are developed that enable one to study the properties of the controller of a prime faithful ideal $I$ of the group algebra $kA$ of an Abelian torsion-free group $A$ of finite rank over a field $k$. The main idea is that the quotient ring $kA/I$ by the given ideal $I$ can be embedded as an integral domain $k[A]$ into some field $F$ and the group $A$ becomes a subgroup of the multiplicative group of the field $F$. This allows one to apply certain results of field theory, such as Kummer's theory and the properties of the multiplicative groups of fields, to the study of the integral domain $k[A]$. In turn, the properties of the integral domain $k[A]\cong kA/I$ depend essentially on the properties of the ideal $I$. In particular, by using these methods, an independent proof of the new version of Brookes's theorem on the controllers of prime ideals of the group algebra $kA$ of an Abelian torsion-free group $A$ of finite rank is obtained in the case where the field $k$ has positive characteristic. Bibliography: 19 titles. 
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     title = {On~the controllers of prime ideals of group algebras {of~Abelian} torsion-free groups of~finite rank over a~field of positive},
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A. V. Tushev. On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive. Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1365-1404. http://geodesic.mathdoc.fr/item/SM_2006_197_9_a6/

[1] I. Lambek, Koltsa i moduli, Mir, M., 1971 | MR | MR | Zbl

[2] K. S. Braun, Kogomologii grupp, Nauka, M., 1987 | MR | MR | Zbl

[3] Ch. J. B. Brookes, “Ideals in group rings of soluble groups of finite rank”, Math. Proc. Cambridge Philos. Soc., 97 (1985), 27–49 | DOI | MR | Zbl

[4] J. E. Roseblade, “Prime ideals in group rings of polycyclic groups”, Proc. London Math. Soc. (3), 36:3 (1978), 385–447 | DOI | MR | Zbl

[5] A. V. Tushev, “Neterovy moduli nad abelevymi gruppami konechnogo svobodnogo ranga”, Ukr. matem. zhurn., 43:7–8 (1991), 1042–1048 | MR | Zbl

[6] D. Segal, “On the group rings of Abelian minimax groups”, J. Algebra, 237:1 (2001), 64–94 | DOI | MR | Zbl

[7] A. V. Tushev, “O neterovykh modulyakh nad minimaksnymi abelevymi gruppami”, Ukr. matem. zhurn., 54:7 (2002), 969–980 | MR | Zbl

[8] A. V. Tushev, “O primitivnykh predstavleniyakh minimaksnykh nilpotentnykh grupp”, Matem. zametki, 72:1 (2002), 131–144 | MR | Zbl

[9] A. V. Tushev, “Indutsirovannye predstavleniya abelevykh grupp konechnogo ranga”, Ukr. matem. zhurn., 55:9 (2003), 1249–1253 | MR | Zbl

[10] A. V. Tushev, “On deviation in groups”, Illinois J. Math., 47:1/2 (2003), 539–550 | MR | Zbl

[11] S. Leng, Algebra, Mir, M., 1968 | MR | Zbl

[12] N. Burbaki, Algebra: moduli, koltsa, formy, Nauka, M., 1966 | MR | Zbl

[13] V. S. Charin, “O gruppakh avtomorfizmov nilpotentnykh grupp”, Ukr. matem. zhurn., 6:3 (1954), 295–304 | MR | Zbl

[14] P. H. Kropholler, P. A. Linnell, J. A. Moody, “Applications of a new $K$-theoretic theorem to soluble group rings”, Proc. Amer. Math. Soc., 104:3 (1988), 675–684 | DOI | MR | Zbl

[15] D. S. Passman, “Algebraic crossed products”, Group actions on rings, Proc. AMS-IMS-SIAM Summer Res. Conf. (Brunswick/Maine 1984), Contemp. Math., 43, 1985, 209–225 | MR | Zbl

[16] D. S. Passman, The algebraic structure of group rings, Wiley, New York, 1977 | MR | Zbl

[17] P. Samyuel, O. Zarisskii, Kommutativnaya algebra, t. 1, 2, IL, M., 1963 | MR | MR | Zbl

[18] G. M. Bergman, “The logarithmic limit-set of an algebraic variety”, Trans. Amer. Math. Soc., 157 (1971), 459–469 | DOI | MR | Zbl

[19] M. Atya, I. Makdonald, Vvedenie v kommutativnuyu algebru, Mir, M., 1972 | MR | MR | Zbl