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Invariant ideals of Abelian group algebras under the multiplicative action of a field. I. (English) Zbl 0992.16021
Let \(F\) be a field and let \(V=F^n\) be a finite-dimensional \(F\)-vector space, viewed multiplicatively. If \(G=F^*\) is the multiplicative group of \(F\), then \(G\) acts on \(V\) and hence on any group algebra \(K[V]\). The goal is to completely describe the semiprime \(G\)-stable ideals of \(K[V]\). Suppose that \(F\) is an infinite locally finite field with \(\text{char }F\neq\text{char }K\). Then the authors prove that every \(G\)-stable ideal of \(K[V]\) can be written uniquely as a finite irredundant intersection of augmentation ideals \(\omega(A_i;V)\), where each \(A_i\) is the kernel of the natural epimorphism \(K[V]\to K[V/A_i]\). Moreover, the set of these \(G\)-invariant ideals is Noetherian. As observed by the authors, this result is essentially the locally-finite analog of a result on the field of rational numbers given by C. J. B. Brookes and D. M. Evans [Math. Proc. Camb. Philos. Soc. 130, No. 2, 287-294 (2001; Zbl 1005.20005)].

MSC:
16S34 Group rings
16D25 Ideals in associative algebras
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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[1] C. J. B. Brookes and D. M. Evans, Augmentation modules for affine groups, Math. Proc. Cambridge Philos. Soc. 130 (2001), 287-294. CMP 2001:06 · Zbl 1005.20005
[2] B. Hartley and A. E. Zalesskiĭ\kern.15em, Group rings of periodic linear groups, unpublished note (1995).
[3] Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. · Zbl 0368.16003
[4] D. S. Passman and A. E. Zalesskiĭ\kern.15em, Invariant ideals of abelian group algebras and representations of groups of Lie type, Trans. AMS 353 (2001), 2971-2982. · Zbl 0981.16024
[5] A. E. Zalesskiĭ, Group rings of simple locally finite groups, Finite and locally finite groups (Istanbul, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, pp. 219 – 246. · Zbl 0839.16021 · doi:10.1007/978-94-011-0329-9_9 · doi.org
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