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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 $$V$$.

##### 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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##### References:
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