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Invariant ideals of Abelian group algebras under the multiplicative action of a field. II. (English) Zbl 0992.16022
This paper is the second part of a series [D. S. Passman and A. E. Zalesskij, Proc. Am. Math. Soc. 130, No. 4, 939-949 (2002; see the preceding review Zbl 0992.16021)].
Let \(D\) be a division ring and let \(V\) be a finite-dimensional right \(D\)-vector space, viewed multiplicatively. If \(G=D^*\) is the multiplicative group of \(D\), then \(G\) acts on \(V\) and hence on any group algebra \(K[V]\). The main result, which the authors prove here, asserts that every \(G\)-stable semiprime 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 a \(D\)-subspace of \(V\). As a consequence, the set of these \(G\)-stable semiprime ideals is Noetherian. Moreover, if \(V\) is a right \(D\)-vector space of arbitrary dimension, then every \(G\)-stable semiprime ideal of \(K[V]\) is an intersection of augmentation ideals \(\omega(A_i;V)\), where again each \(A_i\) is a \(D\)-subspace of \(V\).

16S34 Group rings
16D25 Ideals in associative algebras
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
Full Text: DOI
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[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. Zalesskiń≠\kern.15em, Invariant ideals of abelian group algebras under the multiplicative action of a field, I, Proc. AMS, to appear.
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