Spectra of conjugated ideals in group algebras of abelian groups of finite rank and control theorems

Tushev, Anatolii V.

Glasgow mathematical journal, Tome 38 (1996) no. 3, pp. 309-320

Résumé

Throughout kwill denote a field. If a group Γ acts on aset A we say an element is Γ-orbital if its orbit is finite and write ΔΓ(A) for the subset of such elements. Let I be anideal of a group algebra kA; we denote by I+ the normal subgrou(I+1)∩A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let I be an ideal of a commutative ring K; then the spectrum Sp(I) of I is the set of all prime ideals P of K such that I≤P. If R is a ring, M is an R-module and x ɛ M we denote by the annihilator of x in R. We recall that a group Γ is said to have finite torsion-free rank if it has a finite series in which each factoris either infinite cyclic or locally finite; its torsion-free rank r0(Γ) is then defined to be the number of infinite cyclic factors in such a series.

Détail

DOI : 10.1017/S0017089500031736

Bibliographie

[1] 1.Brookes, C. J. B., Ideals in group rings of soluble groups of finite rank, Math. Proc. Camb. Phil. Soc. 97 (1985), 27–49. Google Scholar | DOI

[2] 2.Brookes, C. J. B. and Brown, K. A., Primitive group rings and Noetherian rings of quotients. Trans. Amer. Math. Soc. 288 (1985), 605–623. Google Scholar | DOI

[3] 3.Brookes, C. J. B. and Brown, K. A., Injective modules, induction maps and endomorphism rings. Proc. London Math. Soc. (3) 67 (1993), 127–158. Google Scholar | DOI

[4] 4.Brown, K. A., The Nullstellensatz for certain group algebras. J. London Math. Soc. 26 (1982) 425–434. Google Scholar | DOI

[5] 5.Hall, P., Finiteness conditions for soluble groups. Proc. London Math. Soc. 4 (1954) 419–436. Google Scholar

[6] 6.Hall, P., On the finiteness of certain soluble groups. Proc. London Math. Soc. 9 (1959) 595–622. Google Scholar | DOI

[7] 7.Harper, D. L., Primitive irreducible representation of nilpotent groups. Math. Proc. Camb. Phil. Soc. 82 (1977), 241–247. Google Scholar | DOI

[8] 8.Harper, D. L., Primitivity in representations of polycyclic groups. Math. Proc. Camb. Phil. Soc. 88 (1980), 15–31. Google Scholar | DOI

[9] 9.Hartley, B., A dual approach to Cernikov modules. Math. Proc. Camb. Math. Soc. 82 (1977), 215–239. Google Scholar | DOI

[10] 10.Musson, I. M., Representations of infinite soluble groups. Glasgow Math. J. 24 (1983), 43–52 Google Scholar | DOI

[11] 11.Musson, I. M., Irreducible modules for polycyclic groupalgebras. Canad. J. Math. 33 (1981), 901–914. Google Scholar | DOI

[12] 12.Nabney, I. T., Soluble minimax groups and their representations. Ph.D. thesis (University of Cambridge, 1989). Google Scholar

[13] 13.Passman, D. S., Infinite crossed products (Academic Press, Boston, 1989). Google Scholar

[14] 14.Roseblade, J. E., Group rings of polycyclic groups, J. Pure Appl. Algebra. 3 (1973) 307–328. Google Scholar | DOI

[15] 15.Segal, D., Irreducible representations of finitely generated nilpotent groups, Math. Proc. Camb. Phil. Soc. 81 (1977), 201–208. Google Scholar | DOI

[16] 16.Wehrfritz, B. A. F., Infinite linear groups (Springer-Verlag, 1973). Google Scholar | DOI

[17] 17.Wehrfritz, B. A. F., Invariant maximal ideals in certain group algebras. J. London Math. Soc. 46 (1992),101–110. Google Scholar | DOI

[18] 18.Wilson, J. S., Soluble products of minimax groups, and nearly surjective derivations. J. Pure and Appl. Algebra. 53 (1988), 297–318. Google Scholar | DOI