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On Group Rings

  • D. B. Coleman (a1)

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Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced by

The first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

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References

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1. Amitsur, S. A., On the semi-simplicity of group algebras, Michigan Math. J. 6 (1959), 251253.
2. Coleman, D. B., Finite groups with isomorphic group algebras, Trans. Amer. Math. Soc. 105 (1962), 18.
3. Connell, I. G., On the group ring, Can. J. Math. 15 (1963), 650685.
4. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley (Interscience), New York, 1962).
5. Jacobson, N., Structure of rings, Amer. Math. Soc. Colloq. Publ., Vol. 37 (Amer. Math. Soc, Providence, R.I., 1964).
6. Passman, D. S., On the semi-simplicity of modular group algebras, Proc. Amer. Math. Soc. 20 (1969), 515519.
7. Passman, D. S., On the semisimplicity of modular group algebras. II, Can. J. Math. 21 (1969), 1137—1145.
8. Perlis, P. and Walker, G. L., Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950), 420426.
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On Group Rings

  • D. B. Coleman (a1)

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