×

Bimodule structure of central simple algebras. (English) Zbl 1378.16030

Let \(F\) be a field and \(A\) be a central simple algebra over \(F\). Let \(K\) a maximal separable field extension of \(F\) inside \(A\). It is a classical fact that \(A=K a K\) for some choice of \(a\) in \(A \setminus K\). However, this is not true for an arbitrary choice of \(a\). For example, if \(K\) is Galois over \(F\), and conjugation of elements in \(K\) by \(a\) induces an automorphism of \(K\), then \(K a K\) is equal to \(K a\).
The goal of this paper is to study the subspaces \(K a K\) of \(A\) and their powers \((K a K)^m\), for different choices of \(a\). The main result (Theorem 15) is a semi-ring isomorphism between \(K\)-\(K\)-sub-bimodules of \(A\) and \(H\)-\(H\)-sub-bisets of \(\text{Gal}(L/F)\) where \(L\) is the Galois closure of the separable field extension \(K/F\), and \(H=\text{Gal}(L/K)\). One of the applications (Proposition 30) is that if \(K\) is cyclic, which implies that \(A\) is generated by \(K\) and an element \(y\) with \(y^n=\beta\) and \(y \ell y^{-1}=\ell^{\sigma}\), where \(\sigma\) generates \(\text{Gal}(K/F)\) and \([K:F]=n\), then every subalgebra of \(A\) of the form \(K a K\) is in fact \(K[y^d]\) for some \(d|n\).
The appendix contains an alternative proof for the classical result mentioned above, that \(A=K a K\) for some choice of \(a \in A \setminus K\).

MSC:

16K20 Finite-dimensional division rings
12E15 Skew fields, division rings
16D20 Bimodules in associative algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albert, A. A., Two element generation of a separable algebra, Bull. Amer. Math. Soc., 50, 786-788 (1944) · Zbl 0061.05501
[2] Guralnick, R., Some applications of subgroup structure to probabilistic generation and covers of curves, (Algebraic Groups and Their Representations. Algebraic Groups and Their Representations, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., vol. 517 (1998)), 301-320 · Zbl 0973.20011
[3] Jacobson, N., Structure of Rings, Amer. Math. Soc. Colloq. Publ., vol. 37 (1956)
[4] Jacobson, N., Generation of separable and central simple algebras, J. Math. Pures Appl., 36, 217-227 (1957) · Zbl 0079.05204
[5] Jacobson, N., Brauer factor sets, Noether factor sets, and crossed products, (Emmy Noether in Bryn Marr (1983), Springer-Verlag: Springer-Verlag New York), 1-19 · Zbl 0528.13006
[6] Jacobson, N., Finite-Dimensional Division Algebras over Fields (1996), Springer · Zbl 0874.16002
[7] Passman, D. S., Permutation Groups (2012), Dover · Zbl 1270.20001
[8] Rowen, L. H., Polynomial Identities in Ring Theory, Pure and Applied Mathematics, vol. 84 (1980), Academic Press · Zbl 0461.16001
[9] Rowen, L. H.; Saltman, D., Dihedral algebras are cyclic, Proc. Amer. Math. Soc., 84, 162-164 (1982) · Zbl 0492.16022
[10] Wilf, H., Generatingfunctionology (1994), Academic Press · Zbl 0831.05001
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.