Invariant ideals of Abelian group algebras under the multiplicative action of a field. II (Q2781239)

This paper is the second part of a series [\textit{D. S. Passman} and \textit{A. E. Zalesskij}, Proc. Am. Math. Soc. 130, No. 4, 939-949 (2002; see the preceding review Zbl 0992.16021)].NEWLINENEWLINENEWLINELet \(D\) 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 \(V\).