On a Decomposition Equality in Modular Group Rings
نویسنده
چکیده
Let G be an abelian group such that A 6 G with p-component Ap and B 6 G, and let R be a commutative ring with 1 of prime characteristic p with nil-radical N(R). It is proved that if Ap 6⊆ Bp or N(R) 6= 0, then S(RG) = S(RA)(1 + Ip(RG;B)) ⇐⇒ G = AB and Gp = ApBp. In particular, if Ap 6= 1 or N(R) 6= 0, then S(RG) = S(RA) × (1 + Ip(RG;B)) ⇐⇒ G = A × B. So, the question concerning the validity of this formula is completely exhausted. The main statement encompasses both the results of this type established by the author in (Hokkaido Math. J., 2000) and (Miskolc Math. Notes, 2005). We also point out and eliminate in a concrete situation an error in the proof of a statement due to T. Zh. Mollov on a decomposition formula in commutative modular group rings (Proceedings of the Plovdiv University-Math., 1973).
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