# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [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 CMP 2001:06 · Zbl 1005.20005 [2] B. Hartley and A. E. Zalesskiĭ\kern.15em, Group rings of periodic linear groups, unpublished note (1995).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
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.