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The associated graded ring of an integral group ring. (English) Zbl 0358.16006
Let $$G$$ be an abelian group, $$S(G)$$ the symmetric algebra of $$G$$ and $$gr ZG$$ the graded ring associated to the filtration of the ring $$ZG$$ by powers of the augmentation ideal $$A_G$$. There is a natural homogeneous surjection $$\theta:S(G) \to gr ZG$$, with $$n$$-th component given by $\theta_n(x_1 \overset \wedge\otimes \dots \overset \wedge\otimes x_n) = \prod_{i=1}^n (x_i-1) + A^{n+1}_G.$
Several results on the kernel of $$\theta$$ are obtained; the main result is a determination (in the case of finite $$p$$-groups) of when $$\theta_n$$, is an isomorphism. The discussion is highly technical but interesting.

MSC:
 16S34 Group rings 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 15A69 Multilinear algebra, tensor calculus
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References:
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