Passi, Inder Bir S.; Vermani, Lekh Raj The associated graded ring of an integral group ring. (English) Zbl 0358.16006 Math. Proc. Camb. Philos. Soc. 82, 25-33 (1977). 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. Reviewer: William H. Gustafson (Lubbock) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 4 Documents MSC: 16S34 Group rings 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 15A69 Multilinear algebra, tensor calculus PDF BibTeX XML Cite \textit{I. B. S. Passi} and \textit{L. R. Vermani}, Math. Proc. Camb. Philos. Soc. 82, 25--33 (1977; Zbl 0358.16006) Full Text: DOI References: [1] Vermani, Proc. Cambridge Philos. Soc. 68 pp 285– (1970) [2] DOI: 10.1080/00927877508822085 · Zbl 0326.20007 · doi:10.1080/00927877508822085 [3] Serre, Multiplicit?s. Lecture Notes in Mathematics 11 (1965) [4] DOI: 10.1007/BF02566808 · Zbl 0259.20034 · doi:10.1007/BF02566808 [5] Passi, Proc. Cambridge Philos. Soc. 66 pp 505– (1969) [6] DOI: 10.1016/0021-8693(68)90017-3 · Zbl 0159.31502 · doi:10.1016/0021-8693(68)90017-3 [7] DOI: 10.1016/0022-4049(74)90036-X · Zbl 0301.16011 · doi:10.1016/0022-4049(74)90036-X [8] Sandling, Proc. Cambridge Philos. Soc. 71 pp 25– (1972) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.