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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas (15787303) 119(1)
We study the Auslander ring of a basic left Köthe ring Λ and give a characterization of basic left Köthe rings in terms of their Auslander rings. We also study the functor category Mod((Λ-mod)op) and characterize basic left Köthe rings Λ by using functor categories Mod((Λ-mod)op). As a consequence we show that there exists a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings. © The Author(s) under exclusive licence to The Royal Academy of Sciences, Madrid 2024.
Canadian Journal of Mathematics (0008414X)
In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category A is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when Mod (fp A) is a QF-3 category, and every representable functor in Mod (fp A) has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings Λ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories A that Mod (fp A) is a QF-3 category, and every representable functor in Mod (fp A) has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander's ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and Ringel-Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture. © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society.
Bulletin of the London Mathematical Society (00246093) 57pp. 2018-2032
Let (Formula presented.) be a finite-dimensional algebra. In this paper, we show that there is a natural bijection between cosilting modules in (Formula presented.) and semibricks in (Formula presented.) satisfying some condition. Also this bijection restricts to a bijection between all semibricks in (Formula presented.) and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence (Int. Math. Res. Not. 16 (2020) 4993–5054) between support (Formula presented.) -tilting modules and right finite semibricks. © 2025 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
Proceedings of the Royal Society of Edinburgh Section A: Mathematics (03082105)
Let Λ be an artin algebra and be an n-cluster tilting subcategory of Λ-mod with. From the viewpoint of higher homological algebra, a question that naturally arose in Ebrahimi and Nasr-Isfahani (The completion of d-abelian categories. J. Algebra 645 (2024), 143-163) is when induces an n-cluster tilting subcategory of Λ-Mod. In this article, we answer this question and explore its connection to Iyama's question on the finiteness of n-cluster tilting subcategories of Λ-mod. In fact, our theorem reformulates Iyama's question in terms of the vanishing of Ext and highlights its relation with the rigidity of filtered colimits of. Also, we show that is an n-cluster tilting subcategory of Λ-Mod if and only if is a maximal n-rigid subcategory of Λ-Mod if and only if if and only if is of finite type if and only if. Moreover, we present several equivalent conditions for Iyama's question which shows the relation of Iyama's question with different subjects in representation theory such as purity and covering theory. © The Author(s), 2025.
Applied Categorical Structures (09272852) 32(2)
We study Morita equivalence and Morita duality for rings with local units. We extend Auslander’s results on the theory of Morita equivalence and the Azumaya–Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya–Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules. © The Author(s), under exclusive licence to Springer Nature B.V. 2024.
Journal of Algebra (00218693) 645pp. 143-163
Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d-cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d-abelian category, known to be equivalent to a d-cluster tilting subcategory of an abelian category A. The completion of M, denoted by Ind(M), is defined as the universal completion of M with respect to filtered colimits. We explore Ind(M) and demonstrate its equivalence to the full subcategory Ld(M) of ModM, comprising left d-exact functors. Notably, Ind(M) as a subcategory of [Formula presented] falls short of being a d-cluster tilting subcategory since it satisfies all properties of a d-cluster tilting subcategory except d-rigidity. For a d-cluster tilting subcategory M of mod A, M→ consists of all filtered colimits of objects from M, is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M→ is a d-rigid subcategory remains unanswered. However, if it is indeed d-rigid, it qualifies as a d-cluster tilting subcategory. In the case d=2, employing cotorsion theory, we establish that M→ is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M→ is a d-cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d, we address the problem and present several equivalent conditions for Iyama's question. © 2024 Elsevier Inc.
Journal of Algebra (00218693) 657pp. 675-703
Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if radC∞ vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C. © 2024 Elsevier Inc.
Journal of Algebra (00218693) 627pp. 1-42
We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types An or A˜n is equal to the corresponding lower bound cluster algebra generated by projective cluster variables. © 2023 Elsevier Inc.
Quarterly Journal of Mathematics (00335606) 74(4)pp. 1545-1570
In this paper, we introduce nℤ-abelian and nℤ-exact categories by axiomatizing properties of nℤ-cluster tilting subcategories. We study these categories and show that every nℤ-cluster tilting subcategory of an abelian (resp., exact) category has a natural structure of an nℤ-abelian (resp., nℤ-exact) category. Also, we show that every small nℤ-abelian category arises in this way, and discuss the problem for nℤ-exact categories. © The Author(s) 2023. Published by Oxford University Press. All rights reserved.
International Mathematics Research Notices (10737928) 2022(22)pp. 18186-18203
Let M be a small n-abelian category. We show that the category of finitely presented functors mod-M modulo the subcategory of effaceable functors mod0-M has an n-cluster tilting subcategory, which is equivalent to M. This gives a higher-dimensional version of Auslander’s formula. © The Author(s) 2021. Published by Oxford University Press. All rights reserved.
Journal of Algebra (00218693) 594pp. 54-73
Let Λ be an artin algebra and M be an n-cluster tilting subcategory of mod Λ. We show that M has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of M if and only if every effaceable functor M→Ab has finite length. As a consequence we show that if mod Λ has an n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of Λ. © 2021 Elsevier Inc.
Journal of Algebra and its Applications (02194988) 20(7)
In this paper, we study the category of finitely generated modules over a class of right 4-Nakayama artin algebras. This class of algebras appear naturally in the study of representation-finite artin algebras. First, we give a characterization of right 4-Nakayama artin algebras. Then, we classify finitely generated indecomposable right modules over right 4-Nakayama artin algebras. We also compute almost split sequences for the class of right 4-Nakayama artin algebras. © 2021 World Scientific Publishing Company.
Canadian Journal of Mathematics (0008414X) 73(3)pp. 656-686
A ring is called right Köthe if every right -module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right Köthe rings in terms of their Coxeter valued quivers. We also give a characterization of basic right Köthe rings with radical square zero. Therefore, we give a solution to Köthe's problem in these two cases. © Canadian Mathematical Society 2020.
Journal of the Australian Mathematical Society (14467887) 109(1)pp. 93-111
We show that a directed graph is a finite graph with no sinks if and only if, for each commutative unital ring, the Leavitt path algebra is isomorphic to an algebraic Cuntz-Krieger algebra if and only if the -algebra is unital and. Let be a field and be the group of units of. When rank k×), we show that the Leavitt path algebra is isomorphic to an algebraic Cuntz-Krieger algebra if and only if is unital and. We also show that any unital -algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz-Krieger algebra, is isomorphic to an algebraic Cuntz-Krieger algebra. As a consequence, corners of algebraic Cuntz-Krieger algebras are algebraic Cuntz-Krieger algebras. © 2019 Australian Mathematical Publishing Association Inc.
Journal of Algebra (00218693) 549pp. 177-194
From the viewpoint of higher homological algebra, we introduce pure semisimple n-abelian categories, which are analogs of pure semisimple abelian categories. Let Λ be an Artin algebra and M be an n-cluster tilting subcategory of Mod-Λ. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let m be an n-cluster tilting subcategory of mod-Λ. We show that Add(m) is an n-cluster tilting subcategory of Mod-Λ if and only if m has an additive generator if and only if Mod(m) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras. © 2019 Elsevier Inc.
Journal of Algebra (00218693) 563pp. 352-375
We prove a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categories. More precisely, for a positive integer n and a small n-abelian category M, we show that M is equivalent to a full subcategory of an abelian category L2(M,G), where L2(M,G) is the category of absolutely pure group valued functors over M. We also show that n-kernels and n-cokernels in M are precisely exact sequences of L2(M,G) with terms in M. © 2020 Elsevier Inc.
Algebras and Representation Theory (1386923X) 23(4)pp. 1297-1317
In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra Λ is representation-finite if and only if Λ is right n-Nakayama for some positive integer n. We classify hereditary right n-Nakayama algebras. We also define right n-coNakayama algebras and show that an artin algebra Λ is right n-coNakayama if and only if Λ is left n-Nakayama. We then study right 2-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right 2-Nakayama algebra. We end by classifying finite dimensional right 2-Nakayama algebras in terms of their quivers with relations. © 2019, Springer Nature B.V.
Proceedings of the American Mathematical Society (00029939) 147(7)pp. 2775-2782
Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and let B be one of the associated tilted algebras. We show that the B-modules, ordered from right to left in the Auslander-Reiten quiver of Λ form a maximal forward hom-orthogonal sequence of Λ-modules whose dimension vectors form the c-vectors of a maximal green sequence for Λ. Thus we give a proof of Igusa-Todorov’s conjecture. © 2019 American Mathematical Society.
Journal of Algebra (00218693) 514pp. 25-39
A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules which are homomorphic image of R-R(k). In this paper, we give a characterization of left k-cyclic rings. As a consequence, we give a characterization of left Kothe rings, which is a generalization of Kothe-Cohen-Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left k-cyclic ring. As a corollary, we show that R is Morita equivalent to a basic left Kothe ring if and only if R is an artinian left multiplicity-free top ring. (C) 2018 Elsevier Inc. All rights reserved.
Journal of Algebra (00218693)
A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules which are homomorphic image of RkR. In this paper, we give a characterization of left k-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe–Cohen–Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left k-cyclic ring. As a corollary, we show that R is Morita equivalent to a basic left Köthe ring if and only if R is an artinian left multiplicity-free top ring. © 2018 Elsevier Inc.
Proceedings of the American Mathematical Society (00029939) 146(6)pp. 2409-2416
For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e., gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons’s conjecture. © 2018 Amerian Mathematial Soiety.
Journal of Algebra (00218693) 511pp. 227-248
For any unital commutative ring R and for any graph E, we identify the commutative core of the Leavitt path algebra of E with coefficients in R, which is a maximal commutative subalgebra of the Leavitt path algebra. Furthermore, we are able to characterize injectivity of representations which gives a generalization of the Cuntz–Krieger uniqueness theorem. © 2018 Elsevier Inc.
Communications in Algebra (00927872) 45(6)pp. 2444-2451
A ring R is called a left APP-ring if for each element a∈R, the left annihilator lR(Ra) is right s-unital as an ideal of R or equivalently R∕lR(Ra) is flat as a left R-module. In this paper, we show that for a ring R and derivation δ of R, R is left APP if and only if R is δ-weakly rigid and the differential polynomial ring R[x;δ] is left APP. As a consequence, we see that if R is a left APP-ring, then the nth Weyl algebra over R is left APP. Also we define δ-left APP (resp. p.q.-Baer) rings and we show that R is left APP (resp. p.q.-Baer) if and only if for each derivation δ of R, R is δ-weakly rigid and δ-left APP (resp. p.q.-Baer). Finally we prove that R[x;δ] is left APP (resp. p.q.-Baer) if and only if R is δ-left APP (resp. p.q.-Baer). © 2017, Copyright © Taylor & Francis.
Communications in Algebra (00927872) 45(1)pp. 446-454
Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary and sufficient conditions for a skew polynomial ring R[x;α] and differential polynomial ring R[x;δ] to be 2-primal. We compute the Jacobson radical and the set of unit elements of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. Also we establish the lower bounds on the stable range of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. As an application we show that if R is 2-primal then the nth Weyl algebra over R is 2-primal and in this case (Formula presented.). As a consequence, we extend and unify several known results of [4], [8], [10], [18], [19], and [22]. © 2017, Copyright © Taylor & Francis Group, LLC.
Communications in Algebra (00927872) 45(1)pp. 443-445
In this article, we show that there exists an SCN ring R such that the polynomial ring R[x] is not SCN. This answers a question posed by T. K. Kwak et al. in [2]. © 2017, Copyright © Taylor & Francis Group, LLC.
Proceedings of the American Mathematical Society (00029939) 145(5)pp. 1969-1980
Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra M inside the Kumjian-Pask algebra KPR(Λ). We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of KPR(Λ) is injective if and only if it is injective on M. © 2016 American Mathematical Society.
Communications in Algebra (00927872) 44(3)pp. 1302-1320
Let R be a ring with an endomorphism α and an α-derivation δ. In this article, for a skew-Armendariz ring R we study some properties of skew polynomial ring R[x; α, δ]. In particular, among other results, we show that for an (α, δ)-compatible skew-Armendariz ring R, γ(R[x; α, δ]) = γ(R)[x; α, δ] = Niℓ*(R)[x; α, δ], where γ is a radical in the class of radicals which includes the Wedderburn, lower nil, Levitzky, and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip, and 2-primal property, transfer between R and the skew polynomial ring R[x; α, δ], in case R is (α, δ)-compatible skew-Armendariz. As a consequence we extend and unify several known results. © 2016, Copyright © Taylor & Francis Group, LLC.
Journal of Pure and Applied Algebra (00224049) 220(12)pp. 3948-3965
We prove that the Crisp and Gow's quiver operation on a finite quiver Q produces a new quiver Q′ with fewer vertices, such that the finite dimensional algebras kQ/J2 and kQ′/J2 are singularly equivalent. This operation is a general quiver operation which includes as specific examples some operations which arise naturally in symbolic dynamics (e.g., (elementary) strong shift equivalent, (in–out) splitting, source elimination, etc.). © 2016 Elsevier B.V.
Forum Mathematicum (09337741) 27(6)pp. 3509-3532
For any field K and for a completely arbitrary graph E, we characterize the Leavitt path algebras LK(E) that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists. © by De Gruyter 2015.
Journal of Algebra (00218693) 444pp. 284-296
Let Λ be a representation-finite C-algebra which has Hall polynomials, with the universal cover Λ~ which is a locally bounded directed C-algebra. In this paper we prove that the Z-Lie algebra L(Λ) associated with Λ which is defined by Riedtmann in [17] and the Z-Lie algebra K(Λ) associated with Λ which is defined by Ringel in [19] are isomorphic. As an application we show that if Λ is a representation-finite (generalized) cluster-tilted algebra or representation-finite trivial extension algebra, then K(Λ). ≅. L(Λ). © 2015 Elsevier Inc.
Communications in Algebra (00927872) 43(12)pp. 5113-5120
Let R be a ring with an endomorphism α and an α-derivation δ. In this article, we first compute the Jacobson radical of NI ℤ-graded rings and show that J(S) = Niℓ(S) if and only if (Formula presented.) is a ℤ-graded NI ring and J(S) ∩ S0 is nil. As a corollary we show that, J(R[x; α]) = Niℓ(R[x; α]) if and only if R[x; α] is NI and J(R[x; α]) ∩ R ⊆ Niℓ(R). If R[x, x−1; α] is NI we prove that, J(R[x, x−1; α]) = Niℓ(R[x, x−1; α]) = Niℓ*(R[x, x−1; α]) = Niℓ(R)[x, x−1; α]. We also provide necessary and sufficient conditions for a skew polynomial ring R[x; α, δ] and skew Laurent polynomial ring R[x, x−1; α] to be NI. © 2015, Copyright Taylor & Francis Group, LLC.
Algebras and Representation Theory (1386923X) 17(4)pp. 1155-1161
Let A be a representation finite algebra over finite field k such that the indecomposable A-modules are determined by their dimension vectors and for each M, L ind(A) and N mod(A), either FMN L=0 or F ML N=0. We show that A has Hall polynomials and the rational extension of its Ringel-Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if A is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in mod(A), or representation finite cluster tilted algebra, then A has Hall polynomials and H (A) ℤ Q=C (A)ℤQ. © 2013 Springer Science+Business Media Dordrecht.
Canadian Mathematical Bulletin (00084395) 57(3)pp. 609-613
We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.
Communications in Algebra (00927872) 42(4)pp. 1565-1570
In this note we first show that for a right (resp. left) Ore ring R and an automorphism σ of R, if R is σ-skew McCoy then the classical right (resp. left) quotient ring Q(R) of R is -skew McCoy. This gives a positive answer to the question posed in Başer et al. [1]. We also characterize semiprime right Goldie (von Neumann regular) McCoy (σ-skew McCoy) rings. © 2014 Copyright Taylor and Francis Group, LLC.
Journal of Algebra and its Applications (02194988) 13(3)
Let R be a ring with an endomorphism α and an α-derivation δ. In this note we show that if R is (α, δ)-compatible then R is 2-primal if and only if the Ore extension R[x; α, δ] is 2-primal if and only if Niℓ(R) = Niℓ*(R; α, δ) if and only if Niℓ(R)[x; α, δ] = Niℓ*(R[x; α, δ]) if and only if every minimal (α, δ)-prime ideal of R is completely prime. © World Scientific Publishing Company.
Journal of Algebra (00218693) 415pp. 234-246
It is shown that, under suitable conditions, an Ore extension R[. x ; α] of a Jacobson ring R (i.e., a ring all of whose prime ideals are semiprimitive) with a monomorphism α, will be Jacobson. These conditions are satisfied by the class of rings studied by Pearson, Stephenson and Watters, and by left Noetherian rings, giving a theorem of Goldie and Michler. © 2014 Elsevier Inc.
Communications in Algebra (00927872) 42(3)pp. 1337-1349
In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x-1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N0(R[x; α]) = Niℓ*(R)[x; α] and J(R[x, x-1; α]) = N0(R[x, x-1; α]) = Niℓ*(R)[x, x-1; α]. © 2014 Copyright Taylor and Francis Group, LLC.
Taiwanese Journal of Mathematics (10275487) 18(3)pp. 931-941
In this note we study the ascending chain conditions on principal left (resp. right) ideals of the skew polynomial ring R[x; α, δ]. We give a characterization of skew polynomial rings R[x; α, δ] that are domains and satisfy the ascending chain condition on principal left (resp. right) ideals. We also prove that if R is an α-rigid ring that satisfies the ascending chain condition on right annihilators and ascending chain condition on principal right (resp. left) ideals, then the skew polynomial ring R[x; α, δ] and skew power series ring R[[x; α]] also satisfy the ascending chain condition on principal right (resp. left) ideals.
Journal of Algebra and its Applications (02194988) 12(1)
Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Niℓ*(R)[[S, ω]], where α is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip and 2-primal property, transfer between R and the skew generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz. © 2013 World Scientific Publishing Company.
Journal of Algebra and its Applications (02194988) 11(4)
For a ring derivation δ, we introduce and investigate a generalization of reduced rings and Armendariz rings which we call a δ-Armendariz ring. Various classes of δ-Armendariz rings is provided and a number of properties of this generalization are established. Radicals and minimal prime ideals of the differential polynomial ring R[x; δ], in terms of those of a δ-Armendariz R, is determined. We prove that several properties transfer between R and the differential polynomial ring R[x; δ], in case R is δ-Armendariz. © 2012 World Scientific Publishing Company.
Algebras and Representation Theory (1386923X) 15(3)pp. 483-490
Let A be a representation finite algebra over finite field k. In this note we first show that the existence of Hall polynomials for A equivalent to the existence of the Hall polynomial φ NL M for each M, L ∈ modA and N ∈ indA. Then we show that for a basic connected Nakayama algebra A, ℋ (A)= ℒ(A) and Hall polynomials exist for this algebra. We also provide another proof of the existence of Hall polynomials for the representation directed split algebras. © 2010 Springer Science+Business Media B.V.
Communications in Algebra (15324125) 39(9)pp. 3114-3132
For a ring endomorphism α, we introduce and investigate SPA-rings which are a generalization of α-rigid rings and determine the radicals of the skew polynomial rings R[x;α], R[x, x-1; α] and the skew power series rings R[[x; α]], R[[x, x-1; α]], in terms of those of R. We prove that several properties transfer between R and the extensions, in case R is an SPA-ring. We will construct various types of nonreduced SPA-rings and show SPA is a strictly stronger condition than α-rigid. © Taylor & Francis Group, LLC.
Communications in Algebra (15324125) 39(11)pp. 4461-4469
For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T n(R, α). By using an ideal theory of a skew triangular matrix ring T n(R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/〈x n〉 for each positive integer n, where R[x; α] is the skew polynomial ring, and 〈x n〉 is the ideal generated by x n. © 2011 Copyright Taylor and Francis Group, LLC.
Bulletin of the Australian Mathematical Society (17551633) 84(3)pp. 455-457
In this note we show that there exist a semiprime ring R, a strictly ordered artinian, narrow, unique product monoid (S,≤) and a monoid homomorphism ω:S→End(R) such that the skew generalized power series ring R[[S,ω]] is semicommutative but R[[S,ω]] is not reversible. This answers a question posed in Marks et al. ['A unified approach to various generalizations of Armendariz rings', Bull. Aust. Math. Soc. 81 (2010), 361-397]. © Copyright Australian Mathematical Publishing Association Inc. 2011.
Communications in Algebra (15324125) 38(2)pp. 567-575
For a ring R we study an ideal theory of a triangular matrix ring and use it to determine radicals and prime ideals of the ring R[x]/〈xn〉, for each positive integer n, where R[x] is the ring of polynomials in an indeterminant x, and xn〉 is the ideal generated by xn. © Taylor & Francis Group, LLC.
Communications in Algebra (00927872) 38(1)pp. 1-10
Let R be a ring and α an injective endomorphism of R, which is not assumed to be surjective. Necessary and sufficient conditions are given for all prime ideals in a skew polynomial ring R[x; α] or skew Laurent ring R[x, x-1; α] to be left Goldie. As a consequence, we obtain a generalization of a result of Goldie and Michler. © Taylor & Francis Group, LLC.
Glasgow Mathematical Journal (1469509X) 51(3)pp. 425-440
Let R be a ring with a monomorphism α and an α-derivation δ. We introduce (α,δ)-weakly rigid rings which are a generalisation of α-rigid rings and investigate their properties. Every prime ring R is (α,δ)-weakly rigid for any automorphism α and α-derivation δ. It is proved that for any n, a ring R is (α,δ)-weakly rigid if and only if the n-by-n upper triangular matrix ring Tn(R) is (ᾱ,δ̄ )-weakly rigid if and only if Mn(R) is (ᾱ,δ̄)-weakly rigid. Moreover, various classes of (α,δ )-weakly rigid rings is constructed, and several known results are extended. We show that for an (α,δ )-weakly rigid ring R, and the extensions R[x], R[[x]], R[x;α,δ ], R[x, x-1; ], R[[x;α ]], R[[x, x -1;α ]], the ring R is quasi-Baer if and only if the extension over R is quasi-Baer. It is also proved that for an (α,δ)-weakly rigid ring R, if any one of the rings R, R[x], R[x;α,δ ] and R[x, x-1;α ] is left principally quasi-Baer, then so are the other three. Examples to illustrate and delimit the theory are provided. © 2009 Glasgow Mathematical Journal Trust.
Bulletin of the Korean Mathematical Society (10158634) 46(6)pp. 1041-1050
Let R be a ring and α a monomorphism of R. We study the skew Laurent polynomial rings R[x, x-1; α] over an α-skew Armendariz ring R. We show that, if R is an α-skew Armendariz ring, then R is a Baer (resp. p.p.-)ring if and only if R[x, x-1; α] is a Baer (resp. p.p.-) ring. Consequently, if R is an Armendariz ring, then R is a Baer (resp. p.p.-)ring if and only if R[x, x-1] is a Baer (resp. p.p.-)ring. © 2009 The Korean Mathematical Society.
Communications in Algebra (15324125) 36(9)pp. 3533-3542
A ring R with a derivation is called -quasi Baer (resp. quasi-Baer), if the right annihilator of every -ideal (resp. ideal) of R is generated by an idempotent, as a right ideal. We show the left-right symmetry of -(quasi) Baer condition and prove that a ring R is -quasi Baer if and only if R[x;] is quasi Baer if and only if R[x;] is [image omitted]-quasi Baer for every extended derivation [image omitted] of . When R is a ring with IFP, then R is -Baer if and only if R[x;] is Baer if and only if R[x;] is [image omitted]-Baer for every extended derivation [image omitted] of . A rich source of examples for -(quasi) Baer rings is provided.
Journal of Algebra and its Applications (02194988) 7(2)pp. 211-224
A ring R is called (right principally) quasi-Baer if the right annihilator of every (principal right) ideal of R is generated by an idempotent. We study on the relationship between the quasi-Baer and p.q.-Baer property of a ring R and these of the Ore extension R[x;α,δ] for any automorphism α and α-derivation δ of R. © 2008 World Scientific Publishing Company.
Communications in Algebra (15324125) 36(2)pp. 508-522
Let be an endomorphism and an -derivation of a ring R. We introduce the notion of skew-Armendariz rings which are a generalization of -skew Armendariz rings and -rigid rings and extend the classes of non reduced skew-Armendariz rings. Some properties of this generalization are established, and connections of properties of a skew-Armendariz ring R with those of the Ore extension R[x; , ] are investigated. As a consequence we extend and unify several known results related to Armendariz rings.
Studia Scientiarum Mathematicarum Hungarica (15882896) 45(4)pp. 469-481
A ring R is called right principally quasi-Baer (or simply right p.q.-Baer) if the right annihilator of a principal right ideal of R is generated by an idempotent. Let R be a ring such that all left semicentral idempotents are central. Let α be an endomorphism of R which is not assumed to be surjective and R be α-compatible. It is shown that the skew power series ring R[[x; α]] is right p.q.-Baer if and only if the skew Laurent power series ring R[[x, x-1; α]] is right p.q.-Baer if and only if R is right p.q.-Baer and any countable family of idempotents in R has a generalized join in I(R). An example showing that the α-compatible condition on R is not superfluous, is provided. © 2008 Akadémiai Kiadó.
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