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The Coefficient Ring of a Primitive Group Ring

  • John Lawrence (a1)

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All rings are associative with unity. A ring R is prime if xRy ≠ 0 whenever x and y are nonzero. A ring R is (left) primitive if there exists a faithful irreducible left R-module.

If the group ring R[G] is primitive, what can we say about R? First, since every primitive ring is prime, we know that R is prime, by the following

THEOREM 1 (Connell [1, 675]). The group ring R[G] is prime if and only if R is prime and G has no non-trivial finite normal subgroup.

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References

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1. Connell, I., On the group ring, Can. J. Math. 15 (1963), 650685.
2. Fisher, J. and Snider, R., Prime von Neumann regular rings and primitive group algebras, Proc. Amer. Math. Soc. U (1974), 244250.
3. Formanek, E., Group rings of free products are primitive, J. Algebra 26 (1973), 508511.
4. Goodearl, K., Prime ideals in regular self-infective rings, Can. J. Math. 25 (1973), 829839.
5. Goodearl, K. and Handelman, D., Simple self-infective rings (to appear).
6. Goodearl, K., Handelman, D., and Lawrence, J., Strongly prime and completely torsion-free rings, Carleton Math. Series.
7. Handelman, D. and Lawrence, J., Strongly prime rings, Trans. Amer. Math. Soc. (to appear).
8. Kaplansky, I., Algebraic and analytic aspects of operator algebras (Amer. Math. Soc).
9. Lambek, J., Lectures on rings and modules (Ginn and Blaisdell, New York, 1966).
10. Osofsky, B., A non-trivial ring with non-rational infective hull, Can. Math. Bull. 10 (1967), 275282.
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The Coefficient Ring of a Primitive Group Ring

  • John Lawrence (a1)

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