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Journal articles on the topic 'Projective modules (Algebra)'

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Published: 4 June 2021

Last updated: 31 July 2024

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1

FRISK, ANDERS, and VOLODYMYR MAZORCHUK. "PROPERLY STRATIFIED ALGEBRAS AND TILTING." Proceedings of the London Mathematical Society 92, no. 1 (2005): 29–61. http://dx.doi.org/10.1017/s0024611505015431.

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We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

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2

Asefa, Dadi. "Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras." Journal of Mathematics 2021 (November 19, 2021): 1–8. http://dx.doi.org/10.1155/2021/8127282.

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Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

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3

Li, Huanhuan. "The Projective Leavitt Complex." Proceedings of the Edinburgh Mathematical Society 61, no. 4 (2018): 1155–77. http://dx.doi.org/10.1017/s001309151800007x.

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AbstractFor a finite quiverQwithout sources, we consider the corresponding radical square zero algebraA. We construct an explicit compact generator for the homotopy category of acyclic complexes of projectiveA-modules. We call such a generator the projective Leavitt complex ofQ. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex ofQis quasi-isomorphic to the Leavitt path algebra ofQop. Here,Qopis the opposite quiver ofQ, and the Leavitt path algebra ofQopis naturally${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.

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4

Asefa, Dadi. "Gorenstein-Projective Modules over Morita Rings." Algebra Colloquium 28, no. 03 (2021): 521–32. http://dx.doi.org/10.1142/s1005386721000407.

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Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].

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5

Popescu, Dorin. "Polynomial rings and their projective modules." Nagoya Mathematical Journal 113 (March 1989): 121–28. http://dx.doi.org/10.1017/s0027763000001288.

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Let R be a regular noetherian ring. A central question concerning projective modules over polynomial R-algebras is the following.(1.1) BASS-QUILLEN CONJECTURE ([2] Problem IX, [10]). Every finitely generated projective module P over a polynomial R-algebra R[T], T = (T1,…, Tn) is extended from R, i.e.P≊R[T]⊗R P/(T)P.

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6

Sheu, Albert Jeu-Liang. "Projective Modules Over Quantum Projective Line." International Journal of Mathematics 28, no. 03 (2017): 1750022. http://dx.doi.org/10.1142/s0129167x17500227.

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Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces [Formula: see text] constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra [Formula: see text] realized as a concrete groupoid C*-algebra, and find its [Formula: see text]-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra [Formula: see text], we identify those quantum principal [Formula: see text]-bundles introduced by Hajac and collaborators among the projections classified.

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7

Maulana, Iqbal. "Schanuel's Lemma in P-Poor Modules." Jurnal Matematika "MANTIK" 5, no. 2 (2019): 76–82. http://dx.doi.org/10.15642/mantik.2019.5.2.76-82.

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Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from a ring with identity, rather than a field. In module theory there is a concept about projective module, i.e. a module over ring R in which it is projective module relative to all modules over ring R. Next, there is the fact that every module over ring R is projective module relative to all semisimple modules over ring R. If P is a module over ring R which it’s projective relative only to all semisimple modules over ring R, then P is called p-poor module. In the discussion of the projective module, there is a lemma associated with the equivalence of two modules K1 and K2 provided that there are two projective modules P1 and P2 such that is isomorphic to . That lemma is known as Schanuel’s lemma in projective modules. Because the p-poor module is a special case of the projective module, then in this paper will be discussed about Schanuel’s lemma in p-poor modules

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8

Wang, Ren. "The MCM-approximation of the trivial module over a category algebra." Journal of Algebra and Its Applications 16, no. 06 (2017): 1750109. http://dx.doi.org/10.1142/s0219498817501092.

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For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is a maximal Cohen–Macaulay approximation of the trivial module and is the tensor identity of the stable category of Gorenstein-projective modules over the category algebra. We give conditions on when the trivial module is Gorenstein-projective.

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9

BRZEZIŃSKI, TOMASZ. "DIVERGENCES ON PROJECTIVE MODULES AND NON-COMMUTATIVE INTEGRALS." International Journal of Geometric Methods in Modern Physics 08, no. 04 (2011): 885–96. http://dx.doi.org/10.1142/s0219887811005440.

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A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.

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10

Zhang, Aiping. "Endomorphism algebras of Gorenstein projective modules." Journal of Algebra and Its Applications 17, no. 09 (2018): 1850177. http://dx.doi.org/10.1142/s0219498818501773.

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Let [Formula: see text] be an Artin algebra, [Formula: see text] be a Gorenstein projective [Formula: see text]-module and [Formula: see text]. We give a characterization of modules on [Formula: see text] and show that if [Formula: see text] is [Formula: see text]-representation-finite, then [Formula: see text] is also [Formula: see text]-representation-finite. As an application, we prove if [Formula: see text] is a CM-finite [Formula: see text]-Gorenstein algebra, then [Formula: see text] is a [Formula: see text]-Igusa-Todorov algebra.

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11

Adamović, Dražen, and Gordan Radobolja. "Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero." Communications in Contemporary Mathematics 21, no. 02 (2019): 1850008. http://dx.doi.org/10.1142/s0219199718500086.

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This paper is a continuation of [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg–Virasoro algebra [Formula: see text] at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for [Formula: see text] is presented by combining a bosonic construction of Whittaker modules from [D. Adamović, R. Lu and K. Zhao, Whittaker modules for the affine Lie algebra [Formula: see text], Adv. Math. 289 (2016) 438–479; arXiv:1409.5354] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of [Formula: see text]-modules.

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12

Guo, Yaguo, and Shilin Yang. "Projective class rings of a kind of category of Yetter-Drinfeld modules." AIMS Mathematics 8, no. 5 (2023): 10997–1014. http://dx.doi.org/10.3934/math.2023557.

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<abstract><p>In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed $ 8m $-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations.</p></abstract>

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13

Helemskii, A. Ya. "Extreme Version of Projectivity for Normed Modules Over Sequence Algebras." Canadian Journal of Mathematics 65, no. 3 (2013): 559–74. http://dx.doi.org/10.4153/cjm-2012-006-2.

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AbstractWe define and study the so-called extreme version of the notion of a projective normed module. The relevant definition takes into account the exact value of the norm of the module in question, in contrast with the standard known definition that is formulated in terms of normtopology.After the discussion of the case where our normed algebra A is just C, we concentrate on the case of the next degree of complication, where A is a sequence algebra satisfying some natural conditions. The main results give a full characterization of extremely projective objects within the subcategory of the category of non-degenerate normed A-modules, consisting of the so-called homogeneous modules. We consider two cases, ‘non-complete’ and ‘complete’, and the respective answers turn out to be essentially different.In particular, all Banach non-degenerate homogeneous modules consisting of sequences are extremely projective within the category of Banach non-degenerate homogeneous modules. However, neither of them, provided it is infinite-dimensional, is extremely projective within the category of all normed non-degenerate homogeneous modules. On the other hand, submodules of these modules consisting of finite sequences are extremely projective within the latter category.

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14

Liu, Miantao, Ruixin Li, and Nan Gao. "Morphism Categories of Gorenstein-projective Modules." Algebra Colloquium 25, no. 03 (2018): 377–86. http://dx.doi.org/10.1142/s1005386718000275.

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Let Λ be an algebra of finite Cohen-Macaulay type and Γ its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor(Λ-Gproj) of Gorenstein-projective Λ-modules in terms of the module category Γ-mod by a categorical equivalence. Based on this, we obtain that some factor category of the epimorphism category Epi(Λ-Gproj) is a Frobenius category, and also, we clarify the relations among Mor(Λ-Gproj), Mor(T2Λ-Gproj) and Mor(Δ-Gproj), where T2(Λ) and Δ are respectively the lower triangular matrix algebra and the Morita ring closely related to Λ.

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15

Assem, Ibrahim, and Flávio Ulhoa Coelho. "Complete slices and homological properties of tilted algebras." Glasgow Mathematical Journal 36, no. 3 (1994): 347–54. http://dx.doi.org/10.1017/s0017089500030950.

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It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

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16

Yu, Peng, and Zhaoyong Huang. "Pure-injectivity in the category of Gorenstein projective modules." Journal of Algebra and Its Applications 16, no. 08 (2016): 1750146. http://dx.doi.org/10.1142/s0219498817501468.

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In this paper, we introduce and study (weak) pure-injective Gorenstein projective modules. Let [Formula: see text] be an Artin algebra. We prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules coincides with the intersection of the category of pure-injective left [Formula: see text]-modules and that of Gorenstein projective left [Formula: see text]-modules. Then, we get an equivalent characterization of virtually Gorenstein algebras (being CM-finite). Furthermore, we prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules is enveloping in the category of left [Formula: see text]-modules; and if [Formula: see text] is virtually Gorenstein, then it is precovering in the category of pure-injective left [Formula: see text]-modules.

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17

Chen, Xiao-Wu, Dawei Shen, and Guodong Zhou. "The Gorenstein-projective modules over a monomial algebra." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (2018): 1115–34. http://dx.doi.org/10.1017/s0308210518000185.

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We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

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18

Li, Shen, René Marczinzik, and Shunhua Zhang. "Gorenstein Projective Dimensions of Modules over Minimal Auslander–Gorenstein Algebras." Algebra Colloquium 28, no. 02 (2021): 337–50. http://dx.doi.org/10.1142/s1005386721000262.

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In this article we investigate the relations between the Gorenstein projective dimensions of [Formula: see text]-modules and their socles for [Formula: see text]-minimal Auslander–Gorenstein algebras [Formula: see text]. First we give a description of projective-injective [Formula: see text]-modules in terms of their socles. Then we prove that a [Formula: see text]-module [Formula: see text] has Gorenstein projective dimension at most [Formula: see text] if and only if its socle has Gorenstein projective dimension at most [Formula: see text] if and only if [Formula: see text] is cogenerated by a projective [Formula: see text]-module. Furthermore, we show that [Formula: see text]-minimal Auslander–Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.

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19

BOUCHIBA, SAMIR, and MOSTAFA KHALOUI. "PERIODIC SHORT EXACT SEQUENCES AND PERIODIC PURE-EXACT SEQUENCES." Journal of Algebra and Its Applications 09, no. 06 (2010): 859–70. http://dx.doi.org/10.1142/s021949881000421x.

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Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45–67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules.

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20

Zhang, Xiaojin. "τ-Rigid Modules for Algebras with Radical Square Zero". Algebra Colloquium 28, № 01 (2021): 91–104. http://dx.doi.org/10.1142/s1005386721000092.

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For a radical square zero algebra [Formula: see text] and an indecomposable right [Formula: see text]-module [Formula: see text], when [Formula: see text] is Gorenstein of finite representation type or [Formula: see text] is [Formula: see text]-rigid, [Formula: see text] is [Formula: see text]-rigid if and only if the first two projective terms of a minimal projective resolution of [Formula: see text] have no non-zero direct summands in common. In particular, we determine all [Formula: see text]-tilting modules for Nakayama algebras with radical square zero.

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21

Razzaque, Asima, and Inayatur Rehman. "Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings." Journal of Mathematics 2022 (August 31, 2022): 1–8. http://dx.doi.org/10.1155/2022/2792450.

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The LA-module is a nonassociative structure that extends modules over a nonassociative ring known as left almost rings (LA-rings). Because of peculiar characteristics of LA-ring and its inception into noncommutative and nonassociative theory, drew the attention of many researchers over the last decade. In this study, the ideas of projective and injective LA-modules, LA-vector space, as well as examples and findings, are discussed. We construct a nontrivial example in which it is proved that if the LA-module is not free, then it cannot be a projective LA-module. We also construct free LA-modules, create a split sequence in LA-modules, and show several outcomes that are connected to them. We have proved the projective basis theorem for LA-modules. Also, split sequences in projective and injective LA-modules are discussed with the help of various propositions and theorems.

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22

XI, CHANGCHANG, and DENGMING XU. "THE FINITISTIC DIMENSION CONJECTURE AND RELATIVELY PROJECTIVE MODULES." Communications in Contemporary Mathematics 15, no. 02 (2013): 1350004. http://dx.doi.org/10.1142/s0219199713500041.

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The famous finitistic dimension conjecture says that every finite-dimensional 𝕂-algebra over a field 𝕂 should have finite finitistic dimension. This conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional 𝕂-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In the paper, we shall work with a more general setting of Artin algebras. Let B be a subalgebra of an Artin algebra A such that the radical of B is a left ideal in A. (1) If the category of all finitely generated (A, B)-projective A-modules is closed under taking A-syzygies, then fin.dim (B) ≤ fin.dim (A) + fin.dim (BA) + 3, where fin.dim (A) denotes the finitistic dimension of A, and where fin.dim (BA) stands for the supremum of the projective dimensions of those direct summands of BA that have finite projective dimension. (2) If the extension B ⊆ A is n-hereditary for a non-negative integer n, then gl.dim (A) ≤ gl.dim (B) + n. Moreover, we show that the finitistic dimension of the trivially twisted extension of two algebras of finite finitistic dimension is again finite. Also, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given. Our approach in this paper is completely different from the one in our earlier papers.

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23

Durğun, Yılmaz. "Subprojectivity domains of pure-projective modules." Journal of Algebra and Its Applications 19, no. 05 (2019): 2050091. http://dx.doi.org/10.1142/s0219498820500917.

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In a recent paper, Holston et al. have defined a module [Formula: see text] to be [Formula: see text]-subprojective if for every epimorphism [Formula: see text] and homomorphism [Formula: see text], there exists a homomorphism [Formula: see text] such that [Formula: see text]. Clearly, every module is subprojective relative to any projective module. For a module [Formula: see text], the subprojectivity domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is [Formula: see text]-subprojective. We consider, for every pure-projective module [Formula: see text], the subprojective domain of [Formula: see text]. We show that the flat modules are the only ones sharing the distinction of being in every single subprojectivity domain of pure-projective modules. Pure-projective modules whose subprojectivity domain is as small as possible will be called pure-projective indigent (pp-indigent). Properties of subprojectivity domains of pure-projective modules and of pp-indigent modules are studied. For various classes of modules (such as simple, cyclic, finitely generated and singular), necessary and sufficient conditions for the existence of pp-indigent modules of those types are studied. We characterize the structure of a Noetherian ring over which every (simple, cyclic, finitely generated) pure-projective module is projective or pp-indigent. Furthermore, finitely generated pp-indigent modules on commutative Noetherian hereditary rings are characterized.

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ASSEM, IBRAHIM, and FLÁVIO U. COELHO. "ENDOMORPHISM ALGEBRAS OF PROJECTIVE MODULES OVER LAURA ALGEBRAS." Journal of Algebra and Its Applications 03, no. 01 (2004): 49–60. http://dx.doi.org/10.1142/s0219498804000691.

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Let A be a connected artin algebra, and e be an idempotent in A such that B=eAe is connected. We show here that if A is laura, left (or right) glued or weakly shod, so is B, respectively. Our proof yields also similar (and known) results for shod and quasi-tilted algebras.

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25

Wu, Dejun, and Hui Zhou. "Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings." Algebra Colloquium 29, no. 03 (2022): 475–90. http://dx.doi.org/10.1142/s1005386722000360.

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Let [Formula: see text] and [Formula: see text] be rings and [Formula: see text] a [Formula: see text]-bimodule. If [Formula: see text] is flat and [Formula: see text] is finitely generated projective (resp., [Formula: see text] is finitely generated projective and [Formula: see text] is flat), then the characterizations of level modules and Gorenstein AC-projective modules (resp., absolutely clean modules and Gorenstein AC-injective modules) over the formal triangular matrix ring [Formula: see text] are given. As applications, it is proved that every Gorenstein AC-projective left [Formula: see text]-module is projective if and only if each Gorenstein AC-projective left [Formula: see text]-module and [Formula: see text]-module is projective, and every Gorenstein AC-injective left [Formula: see text]-module is injective if and only if each Gorenstein AC-injective left [Formula: see text]-module and [Formula: see text]-module is injective. Moreover, Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring [Formula: see text] are studied.

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26

Wang, Yu, and Dexu Zhou. "Gorenstein FC-projective modules." Journal of Algebra and Its Applications 19, no. 04 (2019): 2050066. http://dx.doi.org/10.1142/s0219498820500668.

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In this paper, we investigate Gorenstein FC-projective modules and Gorenstein FC-projective dimensions, and characterize rings over which every module is Gorenstein FC-projective and rings over which every submodule of a projective is Gorenstein FC-projective, respectively. Under an almost excellent extension of rings, we obtain several invariant properties of Gorenstein FC-projective modules and Gorenstein FC-projective dimensions.

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27

PIRKOVSKII, A. YU. "Approximate characterizations of projectivity and injectivity for Banach modules." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 2 (2007): 375–85. http://dx.doi.org/10.1017/s0305004107000163.

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AbstractWe characterize projective and injective Banach modules in approximate terms, generalizing thereby a characterization of contractible Banach algebras given by F. Ghahramani and R. J. Loy. As a corollary, we show that each uniformly approximately amenable Banach algebra is amenable. Some applications to homological dimensions of Banach modules and algebras are also given.

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28

Rieffel, Marc A. "Projective Modules over Higher-Dimensional Non-Commutative Tori." Canadian Journal of Mathematics 40, no. 2 (1988): 257–338. http://dx.doi.org/10.4153/cjm-1988-012-9.

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The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of continuous complex-valued functions under pointwise multiplication. But C(Tn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of Tn as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].)

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29

Endo, Naoki. "On Ratliff-Rush closure of modules." MATHEMATICA SCANDINAVICA 126, no. 2 (2020): 170–88. http://dx.doi.org/10.7146/math.scand.a-119672.

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In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.

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30

Marcos, Eduardo N., Octavio Mendoza, Corina Sáenz та Valente Santiago. "Wide subcategories of finitely generated Λ-modules". Journal of Algebra and Its Applications 17, № 05 (2018): 1850082. http://dx.doi.org/10.1142/s0219498818500822.

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We explore some properties of wide subcategories of the category [Formula: see text] of finitely generated left [Formula: see text]-modules, for some artin algebra [Formula: see text] In particular we look at wide finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras. Furthermore, for a wide class [Formula: see text] in [Formula: see text] we give necessary and sufficient conditions to see that [Formula: see text] for some projective [Formula: see text]-module [Formula: see text] and finally, a connection with ring epimorphisms is given.

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31

Almahdi, Fuad Ali Ahmed, and Mohammed Tamekkante. "Ding w-Flat Modules and Dimensions." Algebra Colloquium 25, no. 02 (2018): 203–16. http://dx.doi.org/10.1142/s1005386718000147.

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The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-flat module if [Formula: see text] is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules … → P1 → P0 → P0 → P1 → … such that M ≅ Im(P0 → P0) and such that the functor HomR(−, F) leaves the sequence exact whenever F is w-flat. Several wellknown classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension (of modules and rings) and the existence of Ding w-flat precovers are also studied.

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32

Chen, Jianlong. "P-Projective modules." Communications in Algebra 24, no. 3 (1996): 821–31. http://dx.doi.org/10.1080/00927879608825603.

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Jun, Jaiung, Kalina Mincheva, and Louis Rowen. "Projective systemic modules." Journal of Pure and Applied Algebra 224, no. 5 (2020): 106243. http://dx.doi.org/10.1016/j.jpaa.2019.106243.

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Rong, Luo. "IG-projective modules." Journal of Pure and Applied Algebra 218, no. 2 (2014): 252–55. http://dx.doi.org/10.1016/j.jpaa.2013.05.010.

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35

Harada, M. "Almost Projective Modules." Journal of Algebra 159, no. 1 (1993): 150–57. http://dx.doi.org/10.1006/jabr.1993.1151.

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JANSSEN, K., and J. VERCRUYSSE. "MULTIPLIER BI- AND HOPF ALGEBRAS." Journal of Algebra and Its Applications 09, no. 02 (2010): 275–303. http://dx.doi.org/10.1142/s0219498810003926.

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We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

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Mao, Lixin. "Finitely projective modules with respect to a semidualizing module." Journal of Algebra and Its Applications 18, no. 03 (2019): 1950049. http://dx.doi.org/10.1142/s021949881950049x.

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Let [Formula: see text] be a commutative ring. We define and study [Formula: see text]-projective modules with respect to a semidualizing [Formula: see text]-module [Formula: see text], which are called [Formula: see text]–[Formula: see text]-projective modules. As consequences, we characterize several rings such as [Formula: see text]-coherent rings and Artinian rings using [Formula: see text]–[Formula: see text]-projective modules. Some known results are extended.

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XI, CHANGCHANG. "Standardly stratified algebras and cellular algebras." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 1 (2002): 37–53. http://dx.doi.org/10.1017/s0305004102005996.

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Let A be an Artin algebra. Then there are finitely many non-isomorphic simple A-modules. Suppose S1, S2, …, Sn form a complete list of all non-isomorphic simple A-modules and we fix this ordering of simple modules. Let Pi and Qi be the projective cover and the injective envelope of Si respectively. With this order of simple modules we define for each i the standard module Δ(i) to be the maximal quotient of Pi with composition factors Sj with j [les ] i. Let Δ be the set of all these standard modules Δ(i). We denote by [Fscr ](Δ) the subcategory of A-mod whose objects are the modules M which have a Δ-filtration, namely there is a finite chain0 = M0 ⊂ M1 ⊂ M2 ⊂ … ⊂ Mt = Mof submodules of M such that Mi/Mi−1 is isomorphic to a module in Δ for all i. The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard module ∇(i) to be the maximal submodule of Qi with composition factors Sj with j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have also the subcategory [Fscr ](∇) of A-mod whose objects are these modules which have a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ(n) is always projective and ∇(n) is always injective.

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LIU, ZHONGKUI, and XIAOYAN YANG. "GORENSTEIN PROJECTIVE, INJECTIVE AND FLAT MODULES." Journal of the Australian Mathematical Society 87, no. 3 (2009): 395–407. http://dx.doi.org/10.1017/s1446788709000093.

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AbstractIn basic homological algebra, projective, injective and flat modules play an important and fundamental role. In this paper, we discuss some properties of Gorenstein projective, injective and flat modules and study some connections between Gorenstein injective and Gorenstein flat modules. We also investigate some connections between Gorenstein projective, injective and flat modules under change of rings.

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Guo, Yaguo, and Sinlin Yang. "Projective class rings of the category of Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra." Electronic Research Archive 31, no. 8 (2023): 5006–24. http://dx.doi.org/10.3934/era.2023256.

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<abstract><p>In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra $ \mathcal{\bar{A}} $ are construted and classified by Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class ring of the category of Yetter-Drinfeld modules over $ \mathcal{\bar{A}} $ is described explicitly by generators and relations.</p></abstract>

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41

Sheu, Albert Jeu-Liang. "A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups." Canadian Journal of Mathematics 39, no. 2 (1987): 365–427. http://dx.doi.org/10.4153/cjm-1987-018-7.

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In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

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Yang, Gang, Zhongkui Liu, and Li Liang. "Ding Projective and Ding Injective Modules." Algebra Colloquium 20, no. 04 (2013): 601–12. http://dx.doi.org/10.1142/s1005386713000576.

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An R-module M is called Ding projective if there exists an exact sequence ⋯ → P1→ P0→ P0→ P1→ ⋯ of projective R-modules with M= Ker (P0→ P1) such that Hom (-,F) leaves the sequence exact whenever F is a flat R-module. In this paper, we develop some basic properties of such modules. Also, properties of Ding injective modules are discussed.

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Schenkel, Alexander. "Module parallel transports in fuzzy gauge theory." International Journal of Geometric Methods in Modern Physics 11, no. 03 (2014): 1450021. http://dx.doi.org/10.1142/s0219887814500212.

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In this paper, we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a Hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of Hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.

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Yang, Yanjiong, and Xiaoguang Yan. "Strict Mittag–Leffler modules over Gorenstein injective modules." Journal of Algebra and Its Applications 19, no. 03 (2019): 2050050. http://dx.doi.org/10.1142/s0219498820500504.

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In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

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Parra, Rafael, and Juan Rada. "Projective Envelopes of Finitely Generated Modules." Algebra Colloquium 18, spec01 (2011): 801–6. http://dx.doi.org/10.1142/s1005386711000678.

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A well known problem in the theory of envelopes is to characterize those rings having the property that every (finitely generated, finitely presented, simple) module has a projective (pre)envelope. In this paper, we introduce the [Formula: see text]-projective modules for an arbitrary class [Formula: see text] of finitely generated modules, and characterize in this general setting those rings for which every module in [Formula: see text] has a projective preenvelope.

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46

Ringel, Claus Michael, and Pu Zhang. "Gorenstein-projective and semi-Gorenstein-projective modules." Algebra & Number Theory 14, no. 1 (2020): 1–36. http://dx.doi.org/10.2140/ant.2020.14.1.

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Amin, Ismail, Yasser Ibrahim, and Mohamed Yousif. "Rad-Projective and Strongly Rad-Projective Modules." Communications in Algebra 41, no. 6 (2013): 2174–92. http://dx.doi.org/10.1080/00927872.2012.654552.

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48

Iacob, Alina. "Projectively coresolved Gorenstein flat and ding projective modules." Communications in Algebra 48, no. 7 (2020): 2883–93. http://dx.doi.org/10.1080/00927872.2020.1723612.

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49

Iacob, Alina. "Generalized Gorenstein Modules." Algebra Colloquium 29, no. 04 (2022): 651–62. http://dx.doi.org/10.1142/s1005386722000463.

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We introduce a generalization of the Gorenstein injective modules: the Gorenstein [Formula: see text]-injective modules (denoted by [Formula: see text]). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor [Formula: see text], with [Formula: see text] any [Formula: see text]-injective module. Thus, [Formula: see text] is the class of classical Gorenstein injective modules, and [Formula: see text] is the class of Ding injective modules. We prove that over any ring [Formula: see text], for any [Formula: see text], the class [Formula: see text] is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For [Formula: see text] we show that [Formula: see text] (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring [Formula: see text] is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein [Formula: see text]-projectives (denoted by [Formula: see text]). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for any[Formula: see text] the class [Formula: see text] is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.

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50

Mao, Lixin, and Nanqing Ding. "Relative FP-Projective Modules." Communications in Algebra 33, no. 5 (2005): 1587–602. http://dx.doi.org/10.1081/agb-200061047.

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