75 research outputs found
Modules Whose Small Submodules Have Krull Dimension
The main aim of this paper is to show that an AB5*-module whose small
submodules have Krull dimension has a radical having Krull dimension. The proof
uses the notion of dual Goldie dimension.Comment: to appear in the Miskolc conference proceeding 199
On the notion of 'retractable modules' in the context of algebras
This is a survey on the usage of the module theoretic notion of a
"retractable module" in the study of algebras with actions. We explain how
classical results can be interpreted using module theory and end the paper with
some open questions.Comment: dedicated to Patrick and John on the occasion of their 70th birthday
When is a smash product semiprime?
It is an open question whether the smash product of a semisimple Hopf algebra
and a semiprime module algebra is semiprime. In this paper we show that the
smash product of a commutative semiprime module algebra over a semisimple
cosemisimple Hopf algebra is semiprime. In particular we show that the central
-invariant elements of the Martindale ring of quotients of a module algebra
form a von Neumann regular and self-injective ring whenever is semiprime.
For a semiprime Goldie PI -module algebra with central invariants we
show that \AH is semiprime if and only if the -action can be extended to
the classical ring of quotients of if and only if every non-trivial
-stable ideal of contains a non-zero -invariant element. In the last
section we show that the class of strongly semisimple Hopf algebras is closed
under taking Drinfeld twists. Applying some recent results of Etingof and
Gelaki we conclude that every semisimple cosemisimple triangular Hopf algebra
over an algebraically closed field is strongly semisimple.Comment: AMS-LaTex, 14 pages (wrong references cleared
Integrals in Hopf algebras over rings
Integrals in Hopf algebras are an essential tool in studying finite
dimensional Hopf algebras and their action on algebras. Over fields it has been
shown by Sweedler that the existence of integrals in a Hopf algebra is
equivalent to the Hopf algebra being finite dimensional. In this paper we
examine how much of this is true for Hopf algebras over rings. We show that
over any commutative ring R that is not a field there exists a Hopf algebra H
over R containing a non-zero integral but not being finitely generated as
R-module. On the contrary we show that Sweedler's equivalence is still valid
for free Hopf algebras or projective Hopf algebras over integral domains.
Analogously for a left H-module algebra A we study the influence of non-zero
left A#H-linear maps from A to A#H on H being finitely generated as R-module.
Examples and application to separability are given.Comment: 26 page
Endomorphism rings of modules over prime rings
Endomorphism rings of modules appear as the center of a ring, as the fix ring
of ring with group action or as the subring of constants of a derivation. This
note discusses the question whether certain *-prime modules (introduced by
Bican et al.) have a prime endomorphism ring. Several conditions are presented
that guarantee the primness of the endomorphism ring. The contours of a
possible example of a *-prime module whose endomorphism ring is not prime are
traced.Comment: 9 page
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