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Register a new userA cancellation theorem for modules over integral group rings
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Authors
John Nicholson
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A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $mathbb{Z}G$ has SFC provided at most one copy of the quaternions $mathbb{H}$ occurs in the Wedderburn decomposition of the real group ring $mathbb{R}G$. This generalises the Eichler condition in the case of integral group rings.
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A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to Prufer domains. In the present paper we investigate some analogs of these concepts for modules over group rings.
In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. We also provide a revision of a concept of Gr{o}bner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.
For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.
In this article we prove a generalization of the Bloch-Wigner exact sequence over commutative rings with many units. When the ring is a domain, we get a generalization of Suslins Bloch-Wigner exact sequence over infinite fields. Our proof is different and is easier, even in its general form. But nevertheless we use some of Suslins results which relates the Bloch group of the ring to the third homology group of the general linear group of the ring. From there we take an easier path.
Using the Luthar--Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups $Co_1$, $Co_2$ and $Co_3$.
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