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 \(V\).
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 \(V\).
Reviewer: S.V.Mihovski (Plovdiv)
MSC:
| 16S34 | Group rings |
| 16D25 | Ideals in associative algebras |
| 20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
Citations:
Zbl 0992.16021References:
| [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] | Daniel R. Farkas and Robert L. Snider, Simple augmentation modules, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 177, 29 – 42. · Zbl 0802.20006 · doi:10.1093/qmath/45.1.29 |
| [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 under the multiplicative action of a field, I, Proc. AMS, to appear. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
