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On the group ring of a free product with amalgamation

  • Camilla R. Jordan (a1)
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Let G = A*HB be the free product of the groups A and B amalgamating the proper subgroup H and let R be a ring with 1. If H is finite and G is not finitely generated we show that any non-zero ideal I of R(G) intersects non-trivially with the group ring R(M), where M = M(I) is a subgroup of G which is a free product amalgamating a finite normal subgroup. This result compares with A. I. Lichtman's results in [6] but is not a direct generalisation of these.

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References
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1.Brown K. A., The singular ideals of group rings, II, Quart. J. Math. Oxford Ser. 229, (1978), 187197.
2.Burgess W. D., Rings of quotients of group rings; Canad. J. Math, 21 (1969), 865875.
3.Djoković D. Ž. and Tang C. Y., On the Frattini subgroup of the generalized free product with amalgamation, Proc. Amer. Math. Soc. 32 (1972), 2123.
4.Jordan C. R., The Jacobson radical of the group ring of a generalised free product, J. London Math. Soc. (2) 11 (1975), 369376.
5.Jordan C. R., Ph.D. Thesis, University of Leeds, 1975.
6.Lichtman A. I., Ideals in group rings of free products with amalgamations and of HNN extensions, preprint.
7.Magnus W., Karras A. and Solitar D., Combinatorial group theory (Interscience, 1966).
8.Neumann B. H., An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503554.
9.Passman D. S., Infinite group rings (Marcel Dekker, 1971).
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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