By K. T. Arasu, Surinder K. Sehgal (auth.), I. B. S. Passi (eds.)
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Additional info for Algebra: Some Recent Advances
Group Rings and Their Augmentation Ideals. LNM, 715, Springer-Verlag, 1979. S. , Lie solvable group rings, Canad J. , lS, 748-757, 1973. S. , Lie dimension subgroups, Coftlm. Algebra, 3, 59-73, 1975. , Observations on group rings, Comm. Algebra. 5,1119-1162,1977. , Group algebras whose units satisfy a group identity II, Proc. Amer. Math. ,llS,657-662,1997. , Integral group rings with nilpotent unit groups, Canad. J. , 28, 954-960, 1976. , p-adic group rings with nilpotent unit groups, J. pure appl.
Serre) Let R be a ring offinite Krull dimension d and let P be a projective R -module of rank ~ d + 1. Then P has a unimodular element. 16), the above theorem of Serre says that if dimension of R is 1 then every projective R-module (of constant rank) is cancellative. The second point to notice is that a Dedekind domain which is not a UFO will furnish an example which will show that Serre's theorem is best possible for general rings. 1) is also the best possible result even if R is a UFO. Meanwhile let P be a projective R-module which can be decomposed as Q ~ RI.
8. , 1997) Let KG be a group algebra ofa torsion group over an infinite field K ofcharacteristic p > ·0. Then the following are equivalent: (i) U(KG) satisfies a group identity. 34 Ashwani K. S. , a polynomial identity not satisfied by M2(K». (iii) G contains a normal subgroup A such that G / A and A' are finite and G' is a p-group of bounded exponent. (iv) KG(2)(= [KG, KG]KG) is nil of bounded index. (v) (U(KG»' is a p-group of bounded exponent. 2. Unit Groups of Integral Group Rings Throughout this section Z will denote the ring of integers, Z(p) its localization at the prime p and Zp, the ring of p-adic integers.
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