Jerzy Matczuk - Academia.edu (original) (raw)

Papers by Jerzy Matczuk

arXiv (Cornell University), May 8, 2016

In this note we answer the question raised by Han et al. in [3] whether an idempotent isomorphic ... more In this note we answer the question raised by Han et al. in [3] 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.

arXiv (Cornell University), May 31, 2016

Question 3 of [3] asks whether the matrix ring M n (R) is nil clean, for any nil clean ring R. It... more Question 3 of [3] asks whether the matrix ring M n (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 Köthe'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.

Communications in Algebra, 1991

We introduce a concept of centrally closed Hopf module algebras and show that centrally closed Ho... more We introduce a concept of centrally closed Hopf module algebras and show that centrally closed Hopf module algebras inherit all basic properties of centrally closed algebras.

arXiv (Cornell University), Oct 10, 2019

arXiv (Cornell University), Jan 12, 2018

arXiv (Cornell University), Oct 7, 2011

Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn-Jordan ex... more Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn-Jordan extension of R. Some results relating finiteness conditions of R and that of A(R; S) are presented. In particular necessary and sufficient conditions for A(R; S) to be left noetherian, to be left Bézout and to be left principal ideal ring are presented. This also offers a solution to Problem 10 from [6].

arXiv (Cornell University), Aug 16, 2019

The sum-essential graph SR(M) of a left R-module M is a graph whose vertices are all nontrivial s... more The sum-essential graph SR(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 SR(M) and its subgraph PR(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.

Cornell University - arXiv, Jun 22, 2022

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let α be a k-a... more 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 (⋄). 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 S = k[[X]][θ, α] satisfies (⋄) for all k-algebra automorphisms α of k[[X]].

U J-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x ... more U J-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x ∈ J(R). The behavior of U J-rings under various algebraic construction is investigated. In particular, it is shown that the problem of lifting the U J property from a ring R to the polynomial ring R[x] is equivalent to the Köthe's problem for F 2-algebras.

Contemporary Mathematics, 2019

A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a ... more 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].

UJ-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x\i... more 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.

We classify modules and rings with some specific properties of their intersection graphs. In part... more 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.

We answer in negative two of questions posed in [4]. We also establish a new characterization of ... more 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.

Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R i... more Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R is semiprime left Goldie then the same holds for the Ore extension R[x;σ,δ] and both rings have the same left uniform dimension. 1

A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a ... more 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].

We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. T... more We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in [5]. We also provide an easy construction of one sided duo rings.

Contemporary Mathematics, 2019

Let n be an arbitrary natural number. The class of (strongly) n-torsion clean rings is introduced... more 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.

arXiv (Cornell University), May 8, 2016

In this note we answer the question raised by Han et al. in [3] whether an idempotent isomorphic ... more In this note we answer the question raised by Han et al. in [3] 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.

arXiv (Cornell University), May 31, 2016

Question 3 of [3] asks whether the matrix ring M n (R) is nil clean, for any nil clean ring R. It... more Question 3 of [3] asks whether the matrix ring M n (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 Köthe'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.

Communications in Algebra, 1991

We introduce a concept of centrally closed Hopf module algebras and show that centrally closed Ho... more We introduce a concept of centrally closed Hopf module algebras and show that centrally closed Hopf module algebras inherit all basic properties of centrally closed algebras.

arXiv (Cornell University), Oct 10, 2019

arXiv (Cornell University), Jan 12, 2018

arXiv (Cornell University), Oct 7, 2011

Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn-Jordan ex... more Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn-Jordan extension of R. Some results relating finiteness conditions of R and that of A(R; S) are presented. In particular necessary and sufficient conditions for A(R; S) to be left noetherian, to be left Bézout and to be left principal ideal ring are presented. This also offers a solution to Problem 10 from [6].

arXiv (Cornell University), Aug 16, 2019

The sum-essential graph SR(M) of a left R-module M is a graph whose vertices are all nontrivial s... more The sum-essential graph SR(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 SR(M) and its subgraph PR(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.

Cornell University - arXiv, Jun 22, 2022

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let α be a k-a... more 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 (⋄). 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 S = k[[X]][θ, α] satisfies (⋄) for all k-algebra automorphisms α of k[[X]].

U J-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x ... more U J-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x ∈ J(R). The behavior of U J-rings under various algebraic construction is investigated. In particular, it is shown that the problem of lifting the U J property from a ring R to the polynomial ring R[x] is equivalent to the Köthe's problem for F 2-algebras.

Contemporary Mathematics, 2019

A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a ... more 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].

UJ-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x\i... more 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.

We classify modules and rings with some specific properties of their intersection graphs. In part... more 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.

We answer in negative two of questions posed in [4]. We also establish a new characterization of ... more 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.

Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R i... more Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R is semiprime left Goldie then the same holds for the Ore extension R[x;σ,δ] and both rings have the same left uniform dimension. 1

A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a ... more 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].

We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. T... more We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in [5]. We also provide an easy construction of one sided duo rings.

Contemporary Mathematics, 2019

Let n be an arbitrary natural number. The class of (strongly) n-torsion clean rings is introduced... more 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.