# zbMATH — the first resource for mathematics

The left and right dimensions of a skew field over the subfield of invariants. (English) Zbl 06715050
Summary: If $$H$$ is a Hopf algebra and $$A$$ an $$H$$-module algebra without nontrivial $$H$$-stable left or right ideals, then the subalgebra of $$H$$-invariant elements $$A^H$$ is a skew field and $$A$$ may be regarded as a vector space over $$A^H$$ with respect to either left or right multiplications. It is proved in the paper that the left dimension of $$A$$ over $$A^H$$ is equal to the right dimension under the assumptions that $$A$$ is semiprimary and $$\dim H < \infty$$. In the case when $$A$$ is itself a skew field, this answers a question raised by J. Bergen, M. Cohen and D. Fischman.
##### MSC:
 16T05 Hopf algebras and their applications
##### Keywords:
Hopf algebras; Hopf module algebras; skew fields; invariants
Full Text:
##### References:
 [1] Bergen, J.; Cohen, M.; Fischman, D., Irreducible actions and faithful actions of Hopf algebras, Israel J. Math., 72, 5-18, (1990) · Zbl 0753.16017 [2] Cohen, M., Smash products, inner actions and quotient rings, Pacific J. Math., 125, 45-66, (1986) · Zbl 0553.16005 [3] Cohen, M.; Fischman, D.; Montgomery, S., Hopf Galois extensions, smash products, and Morita equivalence, J. Algebra, 133, 351-372, (1990) · Zbl 0706.16023 [4] Cohn, P. M., Quadratic extensions of skew fields, Proc. Lond. Math. Soc., 11, 531-556, (1961) · Zbl 0104.03301 [5] Eryashkin, M. S., Invariants and rings of quotients of H-semiprime H-module algebras satisfying a polynomial identity, Izv. Vyssh. Uchebn. Zaved. Mat., Russian Math. (Iz. VUZ), 60, 18-34, (2016), (in Russian); English translation in · Zbl 1345.16032 [6] Etingof, P., Galois bimodules and integrality of PI comodule algebras over invariants, J. Noncommut. Geom., 9, 567-602, (2015) · Zbl 1329.16028 [7] Jacobson, N., Structure of rings, (1956), Amer. Math. Soc. [8] Montgomery, S., Hopf algebras and their actions on rings, (1993), Amer. Math. Soc. · Zbl 0804.16041 [9] Rowen, L. H., Ring theory, vol. I, (1988), Academic Press [10] Schofield, A. H., Artin’s problem for skew field extensions, Math. Proc. Cambridge Philos. Soc., 97, 1-6, (1985) · Zbl 0574.16008 [11] Skryabin, S., Projectivity and freeness over comodule algebras, Trans. Amer. Math. Soc., 359, 2597-2623, (2007) · Zbl 1123.16032 [12] Skryabin, S.; Van Oystaeyen, F., The Goldie theorem for H-semiprime algebras, J. Algebra, 305, 292-320, (2006) · Zbl 1109.16033 [13] Ştefan, D.; Van Oystaeyen, F., The Wedderburn-malcev theorem for comodule algebras, Comm. Algebra, 27, 3569-3581, (1999) · Zbl 0945.16032
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.