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When is a semilocal group algebra continuous? (English) Zbl 1127.16020

Let \(KG\) be the group algebra of a group \(G\) over a field \(K\) of positive characteristic \(p\). The authors prove that if \(KG\) is either (i) a commutative semilocal group algebra or (ii) a local PI algebra, then \(KG\) is continuous (Theorems 4.1 and 4.2).
Furthermore, if \(G\) is an infinite nilpotent group, then the following statements are equivalent: (1) \(KG\) is semiperfect continuous, (2) \(KG\) is semilocal continuous, (3) \(KG\) is semiperfect CS, (4) \(KG\) is semilocal CS and (5) \(G=P\times H\), where \(P\) is an infinite locally finite \(p\)-group and \(H\) is a finite group such that \(p\) does not divide the order of \(H\) (Theorem 4.3). If \(G\) satisfies the condition (5), then \(KG\) is continuous if and only if \(H\) is Abelian (Proposition 4.1).
Two examples are considered: in the first one a semiperfect PI group algebra is constructed which is not continuous and in the second one it is shown that \(KG\) can be continuous without \(G\) being nilpotent.

MSC:

16S34 Group rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16L30 Noncommutative local and semilocal rings, perfect rings
16D50 Injective modules, self-injective associative rings
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References:

[1] Alahmadi, A. N.; Jain, S. K.; Kanwar, P.; Srivastava, J. B., Group algebras in which the complements are direct summands, Fund. Appl. Math., 11, 3, 3-11 (2005) · Zbl 1110.16024
[2] Beidar, K. I.; Jain, S. K.; Kanwar, P.; Srivastava, J. B., CS matrix rings over local rings, J. Algebra, 264, 1, 251-261 (2003) · Zbl 1029.16017
[3] Beidar, K. I.; Jain, S. K.; Kanwar, P.; Srivastava, J. B., Semilocal CS matrix rings of order >1 over group algebras of solvable groups are selfinjective, J. Algebra, 275, 2, 856-858 (2004) · Zbl 1069.16029
[4] Connell, I. G., On the group ring, Canad. J. Math., 15, 650-685 (1963) · Zbl 0121.03502
[5] Jain, S. K.; Kanwar, P.; Srivastava, J. B., Survey of some recent results on CS-group algebras and open questions, (Advances in Algebra (2003), World Scientific Publishers), 401-408 · Zbl 1047.16016
[6] Lam, T. Y., Lectures on Modules and Rings (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0911.16001
[7] Passman, Donald S., The Algebraic Structure of Group Rings (1977), Wiley-Interscience: Wiley-Interscience New York · Zbl 0368.16003
[8] Polcino Milies, César; Sehgal, Sudarshan K., An introduction to group rings, (Algebras and Applications (2002), Kluwer Academic Publishers) · Zbl 0997.20003
[9] Utumi, Yuzo, On continuous rings and self injective rings, Trans. Amer. Math. Soc., 118, 158-173 (1965) · Zbl 0144.27301
[10] Wehrfritz, B. A.F., Infinite Linear Groups (1973), Springer-Verlag: Springer-Verlag Berlin · Zbl 0261.20038
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