On the Uniqueness of the Coefficient Ring in a Group Ring

Adjaero, Isabelle ; Spiegel, Eugene

Canadian journal of mathematics, Tome 35 (1983) no. 4, pp. 654-673

Résumé

Let R 1 and R 2 be commutative rings with identities, G a group and R 1 G and R 2 G the group ring of G over R 1 and R 2 respectively. The problem that motivates this work is to determine what relations exist between R 1 and R 2 if R 1 G and R 2 G are isomorphic. For example, is the coefficient ring R 1 an invariant of R 1 G? This is not true in general as the following example shows. Let H be a group and If R 1 is a commutative ring with identity and R 2 = R 1 H, then but R 1 needn't be isomorphic to R 2.Several authors have investigated the problem when G = <x>, the infinite cyclic group, partly because of its closeness to R[x], the ring of polynomials over R.

Détail

DOI : 10.4153/CJM-1983-037-0

Bibliographie

[1] 1. Adjaero, I., Uniqueness of the coefficient ring and related problems in group rings, (PH.D. Thesis) University of Connecticut (1980). Google Scholar

[2] 2. Akasaki, T., ldempotent ideals in integral group rings, J. Algebra 23 (1972), 343–346. Google Scholar

[3] 3. Batman, J. M. and Coleman, D. B., Group algebras with nilpotent unit groups, Proc. Amer. Math. Soc. 79 (1968). 448–449. Google Scholar

[4] 4. Burgess, W. D., On semi-perfect group rings. Can. Math. Bull. 12 (1969), 645–652. Google Scholar

[5] 5. Cliff, G. H. and Sehgal, S. K., On the trace of an idempotent in a group ring, Proc. Amer. Math. Soc. 62 (1977), 11–14. Google Scholar

[6] 6. Coleman, D. B., Finite groups with isomorphic group algebras. Trans. Amer. Math. Soc. 105 (1962), 1–8. Google Scholar

[7] 7. Coleman, D. B., On group rings. Can. J. Math. 22 (1970), 249–254. Google Scholar

[8] 8. Connell, I. G., On the group ring, Can. J. Math. 75 (1963), 650–685. Google Scholar

[9] 9. Formanek, E., Idempofents in Noetherian group rings, Can. J. Math. 25 (1973). 366–369. Google Scholar

[10] 10. Grunenfelder, L. and Parmenter, M. M., Isomorphic group rings with non-isomorphic coefficient rings. Can. Math. Bull, (to appear). Google Scholar | DOI

[11] 11. Hampton, C. R. and Passman, D. S., On the semisimplicity of group rings of solvable groups. Transactions of the Amer. Math. Soc. 173 (1972), 289–300. Google Scholar

[12] 12. Higman, G., The units of group rings, Proc. London Math. J. 46 (1940), 231–248. Google Scholar

[13] 13. Hochester, M., Non-uniqueness of coefficient rings in a polynomial ring, Proc. Amer. Math. Soc. 34 (1912), 81–82. Google Scholar

[14] 14. Hungerford, W. T., Algebra (Holt Rinehart and Winston, 1974). Google Scholar

[15] 15. Lantz, D. C., R-automorphisms of RG for G abelian torsion free, Proc. amer. Math. Soc. 61 (1976), 1–6. Google Scholar

[16] 16. May, W., Group algebras over finitely generated rings, J. of Alg. 39 (1976), 483–511. Google Scholar

[17] 17. May, W., Isomorphism of group algebras, J. of Alg. 40 (1976), 10–18. Google Scholar

[18] 18. Parmenter, M. M., Isomorphic group rings, Can. Math. Bull. 18 (1975), 567–576. Google Scholar

[19] 19. Parmenter, M. M., Coefficient rings of isomorphic group rings. Bol. Soc. Bras. Mat. 7 (1976), 59–63. Google Scholar

[20] 20. Parmenter, M. M. and Sehgal, S. K., Uniqueness of the coefficient ting in some group rings. Can. Math. Bull. 76 (1973), 551–555. Google Scholar

[21] 21. Passman, D. S., Nil ideals in group rings, Mich. Math. J. 9 (1962), 375–384. Google Scholar

[22] 22. Passman, D. S., Radicals of group rings, Comm. in Alg. 2 (1974), 295–305. Google Scholar

[23] 23. Passman, D. S., The algebraic structure of group rings (John Wiley &amp; Sons, New York, 1977). Google Scholar

[24] 24. Rudin, W. and Schneider, H., Idempotents in group rings, Duke Math. J. 31 (1964), 585–602. Google Scholar

[25] 25. Sehgal, S. K., Topics in group rings (Marcel Dekker, Inc., New York &amp; Basel, 1978). Google Scholar

[26] 26. Sehgal, S. K., Units in commutative integral group rings. Math. J. Okayama Univ. 14 (1970), 135–138. Google Scholar