arXiv subjects

Jerzy Matczuk

Publications and source records attributed to Jerzy Matczuk.

At least 19 recordsLinked to original sources

arXiv2024

The $X$-semiprimeness of Rings

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring $R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover, $X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied.

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arXiv2024

Property $(\diamond)$ for Ore extensions of small Krull dimension

This paper is a continuation of a project to determine which skew polynomial algebras $S = R[\theta; \alpha]$ satisfy property $(\diamond)$, namely that the injective hull of every simple $S$-module is locally artinian, where $k$ is a field, $R$ is a commutative noetherian $k$-algebra, and $\alpha$ is a $k$-algebra automorphism of $R$. Earlier work (which we review) and further analysis done here leads us to focus on the case where $S$ is a primitive domain and $R$ has Krull dimension 1 and contains an uncountable field. Then we show first that if $|\mathrm{Spec}(R)|$ is infinite then $S$ does not satisfy $(\diamond)$. Secondly we show that when $R = k[X]_{ }$ and $\alpha (X) = qX$ where $q \in k \setminus \{0\}$ is not a root of unity then $S$ does not satisfy $(\diamond)$. This is in complete contrast to our earlier result that, when $R = k[[X]]$ and $\alpha$ is an arbitrary $k$-algebra automorphism of infinite order, $S$ satisfies $(\diamond)$. A number of open questions are stated.

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arXiv2022

Critical and injective modules over skew polynomial rings

Let $R$ be a commutative local $k$-algebra of Krull dimension one, where $k$ is a field. Let $α$ be a $k$-algebra automorphism of $R$, and define $S$ to be the skew polynomial algebra $R[θ; α]$. We offer, under some additional assumptions on $R$, a criterion for $S$ to have injective hulls of all simple $S$-modules locally Artinian - that is, for $S$ to satisfy property $(\diamond)$. It is easy and well known that if $α$ is of finite order, then $S$ has this property, but in order to get the criterion when $α$ has infinite order we found it necessary to classify all cyclic (Krull) critical $S$-modules in this case, a result which may be of independent interest. With the help of the above we show that $\hat{S}=k[[X]][θ, α]$ satisfies $(\diamond)$ for all $k$-algebra automorphisms $α$ of $k[[X]]$.

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arXiv2019

Stable range one for rings with central units

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 classical Krull dimension $0$ or $R$ is an algebra over a field $K$ such that $K$ is uncountable and $R$ has only countably many primitive ideals or $R$ is affine and either $K$ has characteristic $0$ or has infinite transcendental degree over its prime subfield or is algebraically closed. However, the general question remains open.

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arXiv2019

Sum-Essential Graphs of Modules

The sum-essential graph $ \mathcal{S}_R(M) $ of a left $R$-module $M$ is a graph whose vertices are all nontrivial submodules of $M$ and two distinct submodules are adjacent iff their sum is an essential submodule of $M$. Properties of the graph $\mathcal{S}_R(M)$ and its subgraph $\mathcal{P}_R(M)$ induced by vertices which are not essential as submodules of $M$ are investigated. The interplay between module properties of $M$ and properties of those graphs is studied.

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arXiv2019

On Fusible Rings

We answer in negative two of questions posed in [4]. We also establish a new characterization of semiprime left Goldie rings by showing that a semiprime ring R is left Goldie iff it is regular left fusible and has finite left Goldie dimension.

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arXiv2018

n-Torsion Clean Rings

Let n be an arbitrary natural number. The class of (strongly) n-torsion clean rings is introduced and investigated. Abelian n-torsion clean rings are somewhat characterized and a complete characterization of strongly n-torsion clean rings is given in the case when n is odd. Some open questions are posed at the end.

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arXiv2017

On UJ-rings

UJ-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x\in J(R). The behavior of UJ-rings under various algebraic construction is investigated. In particular, it is shown that the problem of lifting the UJ property from a ring R to the polynomial ring R[x] is equivalent to the Kothe's problem for F_2-algebras.

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arXiv2017

On the intersection graphs of modules and rings

We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in [A2]. We also generalize this result to modules, i.e. we get the structure theorem of modules for which their intersection graphs are infinite and contain maximal submodules of finite degree. Furthermore we omit the assumption of maximality of submodules and still get a satisfactory characterization of such modules. In addition we show that, if the intersection graph of a module is infinite but its clique number is finite, then the clique and chromatic numbers of the graph coincide. This fact was known earlier only in some particular cases. It appears that such equality holds also in the complement graph.

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arXiv2017

Simple modules and their essential extensions for skew polynomial rings

Let $R$ be a commutative Noetherian ring and $α$ an automorphism of $R$. This paper addresses the question: when does the skew polynomial ring $S = R[θ; α]$ satisfy the property $(\diamond)$, that for every simple $S$-module $V$ the injective hull $E_S(V)$ of $V$ has all its finitely generated submodules Artinian. The question is largely reduced to the special case where $S$ is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when $R$ is an affine algebra over a field $k$ and $α$ is a $k$-algebra automorphism - in this case $(\diamond)$ holds if and only if all simple $S$-modules are finite dimensional over $k$. This leads to a discussion, involving close study of some families of examples, of when this latter condition holds for affine $k$-algebras $S = R[θ;α]$. The paper ends with a number of open questions.

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arXiv2016

A note on semicentral Idempotents

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.

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arXiv2016

Conjugate (nil) clean rings and Kothe's problem

Question 3 of [3] asks whether the matrix ring Mn(R) is nil clean, for any nil clean ring R. It is shown that positive answer to this question is equivalent to positive solution for Kothe's problem in the class of algebras over the field F_2. Other equivalent problems are also discussed. The classes of conjugate clean and conjugate nil clean rings, which lie strictly between uniquely (nil) clean and (nil) clean rings are introduced and investigated.

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arXiv2012

Ring Endomorphisms with Large Images

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism S of a prime one-sided noetherian ring R is injective whenever the image S (R) contains an essential left ideal L of R. If additionally S(L) = L, then S is an automorphism of R. Examples showing that the assumptions imposed on R can not be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.

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arXiv2011

Finiteness conditions of $S$-Cohn-Jordan Extensions

Let a monoid $S$ act on a ring $R$ by injective endomorphisms and $A=A(R,S)$ denote the $S$-Cohn-Jordan extension of $R$. Some results relating finiteness conditions of $R$ and that of $A$ are presented. In particular necessary and sufficient conditions for $A$ to be left noetherian, to be left Bézout and to be left principal ideal ring are presented.

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arXiv2010

On q-skew Iterated Ore Extensions Satisfying a Polynomial Identity

For iterated Ore extensions satisfying a polynomial identity we present an elementary way of erasing derivations. As a consequence we recover some results obtained by Haynal in "PI degree parity in q-skew polynomial rings" (J. Algebra 319, 2008, 4199-4221). We also prove, under mild assumptions on $R_n=R[x_1;\si_1,\de_1]...[x_n;\si_n;\de_n]$ that the Ore extension $R[x_1;\si_1]...[x_n;\si_n]$ exists and is PI if $R_n$ is PI.

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arXiv2009

A description of quasi-duo Z-graded rings

A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a strongly Z-graded ring is left quasi-duo if and only if it is right quasi-duo. This gives a partial answer to a problem posed by Dugas and Lam in [1].

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arXiv2009

Quasi-Duo Skew Polynomial Rings

A characterization of right (left) quasi-duo skew polynomial rings of endomorphism type and skew Laurent polynomial rings are given. In particular, it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is commutative modulo its Jacobson radical iff R[x] is left quasi-duo, (2) the skew Laurent polynomial ring is right quasi-duo iff it is left quasi-duo. These extend some known results concerning a description of quasi-duo polynomial rings and give a partial answer to the question posed by Lam and Dugas whether right quasi-duo rings are left quasi-duo.

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arXiv2009

Double Ore Extensions versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin-Schelter regular algebras of global dimension four, J.J. Zhang and J. Zhang (2008) introduced an algebra extension $A_P[y_1,y_2;σ,δ,τ]$ of $A$, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over $A$. The aim of this paper is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form $A[y_1;σ_1, δ_1][y_2;σ_2, δ_2]$. We also give partial answers to some questions posed in Zhang and Zhang (2008).

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