Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 395-400 (2001)

Actions of Hopf Algebras on Fully Bounded Noetherian Rings

Thomas Guédénon

21, Avenue de Versailles, 75016 Paris, France; e-mail:

Abstract: Let $k$ be a commutative ring, $H$ a finitely generated projective Hopf algebra over $k$ and $R$ a $k$-algebra which is a left $H$-module algebra. Assume that for every $H$-invariant left ideal $I$ of $R$ and every $x+I\in (R/I)^H$ there exists $s\in R^H$, such that $s-x\in I$. The main result of the paper is that $R$ is left FBN if and only if $R$ is left Noetherian and $R^H$ is left FBN. This result generalizes [D, Theorem 8] and [G, Theorem 2.3]. \item{[D]} Dascalescu, S.; Kelarev, A. V.; Torrecillas, B.: FBN Hopf module algebras. Comm. in Algebra 25(11) (1997), 3521-3529. \item{[G]} Garcia, J. J.; Del Rio, A.: Actions of groups on fully bounded Noetherian rings. Comm. in Algebra 22(5) (1994), 1495-1505.

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