Christian Lomp

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• PhD
• Professor (Associate) at University of Porto

A ring with identity is called essentially right quasi-duo if every essential maximal right ideal of it is a two-sided ideal. Essentially right quasi-duo rings generalize essentially right duo rings, a notion that arose in the study of hypercyclic rings, and right quasi-duo rings, as introduced by S.H. Brown. We prove that a ring R is essentially r...

Hypercyclic rings are those rings over which cyclic modules have cyclic injective hulls. Such rings were introduced by Caldwell in 1966 and studied by him and Osofsky in 1968. In this paper, we revisit Osofsky’s work on hypercyclic rings, highlighting some of her results and questions on the subject.

It is well known that a ring R is right Kasch if each simple right R-module embeds in a projective right R-module. In this paper we study the dual notion and call a ring R right dual Kasch if each simple right R-module is a homomorphic image of an injective right R-module. We prove that R is right dual Kasch if and only if every finitely generated...

We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore ex...

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring R with stable range one and central units is commutative. We show that this is the case under any of the following additional conditions: R is semiprime or R is one-sided Noetherian or R has unit-stable range 1 or R has classic...

Matlis showed that the injective hull of a simple module over a commutative Noetherian ring is Artinian. Many non-commutative Noetherian rings whose injective hulls of simple modules are locally Artinian have been extensively studied recently. This property had been denoted by property $(\diamond)$. In this paper we investigate, which non-Noetheria...

Panov proved necessary and sufficient conditions to extend the Hopf algebra structure of an algebra $R$ to an Ore extension $R[x;\sigma,\delta]$ with $x$ being a skew-primitive element. In this paper we extend Panov's result to Ore extensions over weak Hopf algebras. As an application we study Ore extensions of connected groupoid algebras.

In this note we answer two questions on quasi-Baer modules raised by Lee and Rizvi in J.Algebra (2016).

It is shown that, under some natural assumptions, the tensor product of differentially smooth algebras and the skew-polynomial rings over differentially smooth algebras are differentially smooth.

As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine versions of diagram algebras like affine Hecke algebras of type A and affine Temperley-Lieb algebras in a unifying fashion. Affine cellular algebras include Kleshchev's graded quasihereditary algebras, KLR alg...

In this note we answer the question raised by Han et al. in J. Korean Math. Soc (2014) whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the Dedekind-finite rings. An application to module theory is given.

We analyse the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebra $H$ on the $n$th Weyl algebra $A=A_n(K)$ over a field $K$ of characteristic $0$ factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebra $H$ o...

In [22. Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . Pacific J. Math. 91 : 249 – 261 . View all references] Camillo and Zelmanowitz stated that rings all whose modules are dimension modules are semisimple Artinian. It seem however that the proof in [22. Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . P...

Yassemi’s “second submodules” are dualized and properties of its spectrum are studied. This is done by moving the ring theoretical setting to a lattice theoretical one and by introducing the notion of a (strongly) topological lattice ℒ = (L,∧,∨) with respect to a proper subset X of L. We investigate and characterize (strongly) topological lattices...

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.

This is a survey article on a question, posed in 1985 by M.Cohen, whether the smash product $A\#H$ of a semisimple Hopf algebra and a semiprime left $H$-module algebra $A$ is itself semiprime.

Hom-connections and associated integral forms have been introduced and studied by T.Brzezi\'nski as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus $(\Omega, d)$ over an algebra $A$ yields the integral complex which for various algebras has been shown to be i...

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given...

We study injective hulls of simple modules over differential operator rings, providing sufficient conditions under which these modules are not locally Artinian. As a consequence we characterize Ore extensions $S=K[x][y;\sigma, d]$ such that injective hulls of simple $S$-modules are locally Artinian.

We show that the finite dimensional nilpotent complex Lie superalgebras g whose injective hulls of simple U(g)-modules are locally Artinian are precisely those whose even part g0 is isomorphic to a nilpotent Lie algebra with an abelian ideal of codimension 1 or to a direct product of an abelian Lie algebra and a certain 5-dimensional or a certain 6...

Endomorphism rings of modules appear as the center of a ring, as the fix ring of a ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain ∗-prime modules have a prime endomorphism ring. Several conditions are presented that guarantee the primeness of the endomorphism ring. The contour...

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given...

Based on a lattice-theoretic approach, we give a complete characterization of modules with Fleury's spanning dimension. An example of a non-Artinian, non-hollow module satisfying this finiteness condition is constructed. Furthermore we introduce and characterize the dual notion of Fleury's spanning dimension.

Summary: Following it V. Linchenko and it S. Montgomery's arguments [Proc. Am. Math. Soc. 135, No. 10, 3091-3098 (2007; Zbl 1139.16027)] we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime. The paper is freely available on the publisher's homepage http://w...

Zhou defined δ -semiperfect rings as a proper generalization of semiperfect rings. The purpose of this paper is to discuss relative notions of supplemented modules and to show that the semiperfect rings are precisely the semilocal rings which are δ -supplemented. Module theoretic version of our results are obtained.

The purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian Down-Up algebras. We will show that the Noetherian Down-Up algebras A(α, β, γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α, β, γ)-modules are l...

Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every finitely cogenerated inject...

Any finite set of linear operators on an algebra A yields an operator algebra B and a module structure on A, whose endomorphism ring is isomorphic to a subring AB of certain invariant elements of A. We show that if A is a critically compressible left B-module, then the dimension of its self-injective hull  over the ring of fractions of AB is bound...

A left $R$-module $M$ is called weakly supplemented if for every submodule $N$ of $M$ there exists a submodule $L$ of $M$ with $N+L=M$ and $Ncap L$ small in $M$. The authors show that a ring $R$ is left perfect if and only if every left $R$-module is weakly supplemented if and only if $R$ is semilocal and the radical of the countably infinite free...

Let $R$ be an associative ring endowed with a derivation map $delta$ and $tau$ a hereditary torsion theory on the category Mod-$R$ of right $R$-modules, with associated Gabriel filter $cal F_tau$ and associated torsion radical $t_tau$. Sharpening an earlier result of Golan, it P. E. Bland [J. Pure Appl. Algebra 204, No. 1, 1-8 (2006; Zbl 1102.16020...

From the introduction: it A. Badawi considered [in Lect. Notes Pure Appl. Math. 205, 101-110 (1999; Zbl 0962.13018)] a class of commutative rings $R$, called $varphi$-rings, whose nil radical $textNil(R)$ forms a prime ideal, comparable to any other ideal of $R$ in order to create a framework to study pseudo-valuation rings. This class of rings was...

Given a finite group G acting as automorphisms on a ring A, the skew group ring A*G is an important tool for studying the structure of G-stable ideals of A. The ring A*G is G-graded, i.e.G coacts on A*G. The Cohen-Montgomery duality says that the smash product A*G#k[G]^* of A*G with the dual group ring k[G]^* is isomorphic to the full matrix ring M...

Motivated by the study of von Neumann regular skew groups as carried out by Alfaro, Ara and del Rio in 1995 we investigate regular and biregular Hopf module algebras. If $A$ is an algebra with an action by an affine Hopf algebra $H$, then any $H$-stable left ideal of $A$ is a direct summand if and only if $A^H$ is regular and the invariance functor...

Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors where the colocalised coalgebra turned out to be a suitable subcoalgebra. Rather then aiming at a subcoalgebra we look at possible coalgebra covers p:D->>C that...

It follows from a recent paper by Ding and Wang that any ring which is generalized supplemented as left module over itself is semiperfect. The purpose of this note is to show that Ding and Wang's claim is not true and that the class of generalized supplemented rings lies properly between the class of semilocal and semiperfect rings. Moreover we rec...

Summary: A module $M$ is called hollow-lifting if every submodule $N$ of $M$ such that $M/N$ is hollow contains a direct summand $D \subseteq N$ such that $N/D$ is a small submodule of $M/D$. A module $M$ is called lifting if such a direct summand $D$ exists for every submodule $N$. We construct an indecomposable module $M$ without non-zero hollow...

In this note we show that a ring R is left perfect if and only if every left R-module is weakly supplemented if and only if R is semilocal and the radical of the countably infinite free left R-module has a weak supplement.

We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive and...

Summary: At the beginning of his mathematical career Kostia Beidar was working on rings with polynomial identities and primeness conditions for rings. By Posner's theorem the two-sided quotient ring of a prime PI-ring is a finite matrix ring over some field. This result was extended by Martindale to rings with generalised polynomial identities by t...

Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. There is a certain asymmetry in this duality. While the theory of extending modules is well documented in monographs and text books, the purpose of our monograph is to provide a thorough study of supplements and projec...

Let $k$ be a commutative ring, $H$ a Hopf $k$-algebra, $A$ an $H$-module algebra and $A#H$ the smash product of $A$ and $H$. The author investigates sufficient conditions to extend the $H$-action on $A$ to a localization of $A$ (in particular to $Q_cal F(A)$, the ring of quotients with respect to the Gabriel topology $cal F:=cal F_H$ which has a ba...

Proper classes of monomorphisms and short exact sequences were introduced by Buchsbaum to study relative homo-logical algebra. It was observed in abelian group theory that com-plement submodules induce a proper class of monomorphisms and this observations were extended to modules by Stenström, Gener-alov, and others. In this note we consider comple...

Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. There is a certain asymmetry in this duality. While the theory of extending modules is well documented in monographs and text books, the purpose of our monograph is to provide a thorough study of supplements and projec...

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 r...

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 H-invariant elements of the Martindale ring...

Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules ℒ(M) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which ℒ(M) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz'...

Algebra extensions A⊆B where A is a left B-module such that the B-action extends the multiplication in A are ubiquitous. We encounter examples of such extensions in the study of group actions, group gradings or more general Hopf actions as well as in the study of the bimodule structure of an algebra. In this paper we are extending R.Wisbauer's meth...

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 $H$-invariant elements of the Martindale ri...

Düsseldorf, Universiẗat, Diss., 2002. Computerdatei im Fernzugriff.

Let $R$ be an associative ring with identity element. The authors prove that a projective left $R$-module $P$, whose ring of endomorphisms is semilocal, is finitely generated if and only if the dual Goldie dimensions of $P$ and of the factor module $P/J(P)$ of $P$ by its Jacobson radical $J(P)$ are equal.

Extending the reviewer's result, the author proves that for a semilocal ring $R$ with essential left socle the Goldie torsion theory is centrally splitting if and only if $textsoc(_RR)^2subseteqtextsoc(R_R)$. In order to verify that his result extends the reviewer's one, he also gives an example of a semilocal ring that is not a dual ring.

. The splitting of the Goldie (or singular) torsion theory has been extensively studied. Here we determine an appropriate dual Goldie torsion theory, discuss its splitting and answer in the negative a question proposed by Ozcan and Harmanc as to whether the splitting of the dual Goldie torsion theory implies the ring to be quasi-Frobenius. 1. Intro...

In this note we compare some notions of primeness for modules existing in the literature. We characterize the prime left R-modules which the left annihilator of every element is a (two-sided) ideal of R, where R is an associative ring with unity, and we prove that if M is such a nonzero left R-module then M is strongly prime. This two notions are s...

It is well-known that a ring R is semiperfect if and only if R as a left (or as a right) R-module is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if R as a left (or as a right) R-module is...

The splitting of the Goldie (or singular) torsion theory has been extensively studied. Here we determine an appropriate dual Goldie torsion theory, discuss its splitting and answer in the negative a question proposed by Ozcan and Harmanci as to whether the splitting of the dual Goldie torsion theory implies the ring to be quasi-Frobenius.

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.

We explain some formulae that appeared in bounds on DNA codes using elementary methods in metric spaces.

Following Linchenko and Montgomery's arguments we show that the smash product of a semiprime module algebra, satisfying a polynomial iden-tity and an involutive weak Hopf algebra is semiprime. We get new insight into the existence of non-trivial central invariant elements in non-trivial H-stable ideals of subdirect products of certain H-prime modul...

Bican, Jambor, Kepka and Nemec defined a product on the lat-tice of submodules of a module, making any module into a partially ordered groupoid. Submodules that are idempotent with respect to this product be-have similar as idempotent ideals in rings. In particular jansian torsion theories can be described through idempotent submodules. Moreover so...