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Статті в журналах з теми "Pure-semisimple rings"

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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Wisbauer, Robert. "Semisimple and pure semisimple functor rings." Communications in Algebra 18, no. 7 (January 1, 1990): 2343–54. http://dx.doi.org/10.1080/00927879008824024.

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2

Prest, Mike. "Duality and Pure-Semisimple Rings." Journal of the London Mathematical Society s2-38, no. 3 (December 1988): 403–9. http://dx.doi.org/10.1112/jlms/s2-38.3.403.

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3

Mazari-Armida, Marcos. "Superstability, noetherian rings and pure-semisimple rings." Annals of Pure and Applied Logic 172, no. 3 (March 2021): 102917. http://dx.doi.org/10.1016/j.apal.2020.102917.

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4

DUNG, NGUYEN VIET, and JOSÉ LUIS GARCÍA. "DEFINABLE SUBCATEGORIES OVER PURE SEMISIMPLE RINGS." Journal of Algebra and Its Applications 11, no. 05 (September 26, 2012): 1250099. http://dx.doi.org/10.1142/s0219498812500995.

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Анотація:

Let R be a right pure semisimple ring, and [Formula: see text] be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory [Formula: see text] in Mod-R determined by the family [Formula: see text], and show that [Formula: see text] has several nice properties similar to those of the category Mod-R. For example, its functor category [Formula: see text] is a module category, and preinjective objects of [Formula: see text] are sources of left almost split morphisms in [Formula: see text] and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.

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5

Dung, Nguyen Viet, and José Luis García. "Preinjective modules over pure semisimple rings." Journal of Pure and Applied Algebra 212, no. 5 (May 2008): 1207–21. http://dx.doi.org/10.1016/j.jpaa.2007.09.006.

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6

Dung, Nguyen Viet, and José Luis García. "Endofinite modules and pure semisimple rings." Journal of Algebra 289, no. 2 (July 2005): 574–93. http://dx.doi.org/10.1016/j.jalgebra.2005.01.004.

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7

Fernández-Alonso, Rogelio, and Eder S. Martelo. "Preradicals over left pure semisimple hereditary rings." Communications in Algebra 49, no. 7 (March 11, 2021): 3145–60. http://dx.doi.org/10.1080/00927872.2021.1888963.

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8

Zayed, Maher. "Indecomposable modules over right pure semisimple rings." Monatshefte f�r Mathematik 105, no. 2 (June 1988): 165–70. http://dx.doi.org/10.1007/bf01501169.

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9

Dung, Nguyen Viet, and José Luis García. "Splitting torsion pairs over pure semisimple rings." Journal of Pure and Applied Algebra 219, no. 7 (July 2015): 2637–57. http://dx.doi.org/10.1016/j.jpaa.2014.09.020.

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10

Dung, Nguyen Viet, and José Luis García. "Indecomposable modules over pure semisimple hereditary rings." Journal of Algebra 371 (December 2012): 577–95. http://dx.doi.org/10.1016/j.jalgebra.2012.09.004.

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11

Hu, Jiangsheng, Haiyu Liu, and Yuxian Geng. "When every pure ideal is projective." Journal of Algebra and Its Applications 15, no. 02 (October 6, 2015): 1650030. http://dx.doi.org/10.1142/s0219498816500304.

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Анотація:

In this paper, we study the class of rings in which every pure ideal is projective. We refer to rings with this property as PIP-rings. Some properties and examples of PIP-rings are given. When R is a PIP-ring, some new homological dimensions for complexes are given. As applications, we give some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings.

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12

Angeleri Hügel, Lidia. "A key module over pure-semisimple hereditary rings." Journal of Algebra 307, no. 1 (January 2007): 361–76. http://dx.doi.org/10.1016/j.jalgebra.2006.06.025.

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13

Angeleri Hügel, Lidia, and Dolors Herbera. "Auslander–Reiten components over pure-semisimple hereditary rings." Journal of Algebra 331, no. 1 (April 2011): 285–303. http://dx.doi.org/10.1016/j.jalgebra.2010.10.039.

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14

CÁRCELES, A. I., and J. L. GARCÍA. "PURE SEMISIMPLE FINITELY ACCESSIBLE CATEGORIES AND HERZOG'S CRITERION." Journal of Algebra and Its Applications 06, no. 06 (December 2007): 1001–25. http://dx.doi.org/10.1142/s0219498807002648.

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Анотація:

Let [Formula: see text] be a finitely accessible category with products, and assume that its symmetric category [Formula: see text] is also finitely accessible and pure semisimple. We study necessary and sufficient conditions in both categories for [Formula: see text] (and hence [Formula: see text]) to be of locally finite representation type. In particular, we obtain a generalization of Herzog's criterion for finite representation type of left pure semisimple and right artinian rings. As an application, we prove that a left pure semisimple ring R with enough idempotents which has a self-duality is of locally finite representation type if and only if it is left locally finite.

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15

HERZOG, IVO, and PHILIPP ROTHMALER. "WHEN COTORSION MODULES ARE PURE INJECTIVE." Journal of Mathematical Logic 09, no. 01 (June 2009): 63–102. http://dx.doi.org/10.1142/s0219061309000835.

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We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts.

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16

LÓPEZ-PERMOUTH, S. R., J. MASTROMATTEO, Y. TOLOOEI, and B. UNGOR. "PURE-INJECTIVITY FROM A DIFFERENT PERSPECTIVE." Glasgow Mathematical Journal 60, no. 1 (March 14, 2017): 135–51. http://dx.doi.org/10.1017/s0017089516000616.

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AbstractThe study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

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17

Dung, Nguyen Viet, and Daniel Simson. "The Gabriel–Roiter Measure for Right Pure Semisimple Rings." Algebras and Representation Theory 11, no. 5 (May 21, 2008): 407–24. http://dx.doi.org/10.1007/s10468-008-9084-7.

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18

Demirci, Yilmaz Mehmet. "Modules and abelian groups with a bounded domain of injectivity." Journal of Algebra and Its Applications 17, no. 06 (May 23, 2018): 1850108. http://dx.doi.org/10.1142/s0219498818501086.

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In this work, impecunious modules are introduced as modules whose injectivity domains are contained in the class of all pure-split modules. This notion gives a generalization of both poor modules and pure-injectively poor modules. Properties involving impecunious modules as well as examples that show the relations between impecunious modules, poor modules and pure-injectively poor modules are given. Rings over which every module is impecunious are right pure-semisimple. A commutative ring over which there is a projective semisimple impecunious module is proved to be semisimple artinian. Moreover, the characterization of impecunious abelian groups is given. It states that an abelian group [Formula: see text] is impecunious if and only if for every prime integer [Formula: see text], [Formula: see text] has a direct summand isomorphic to [Formula: see text] for some positive integer [Formula: see text]. Consequently, an example of an impecunious abelian group which is neither poor nor pure-injectively poor is given so that the generalization defined is proper.

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19

Simson, Daniel. "On right pure semisimple hereditary rings and an Artin problem." Journal of Pure and Applied Algebra 104, no. 3 (November 1995): 313–32. http://dx.doi.org/10.1016/0022-4049(94)00068-x.

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20

García, José L. "Small potential counterexamples to the pure semisimplicity conjecture." Journal of Algebra and Its Applications 17, no. 10 (October 2018): 1850183. http://dx.doi.org/10.1142/s0219498818501839.

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Анотація:

The pure semisimplicity conjecture or pssc states that every left pure semisimple ring has finite representation type. Let [Formula: see text] be division rings, and assume we identify conditions on a [Formula: see text]-[Formula: see text]-bimodule [Formula: see text] which are sufficient to make the triangular matrix ring [Formula: see text] into a left pure semisimple ring which is not of finite representation type. It is then said that those conditions yield a potential counterexample to the pssc. Simson [17–20] gave several such conditions in terms of the sequence of the left dimensions of the left dual bimodules of [Formula: see text]. In this paper, conditions with the same purpose are given in terms of the continued fraction attached to [Formula: see text], and also through arithmetical properties of a division ring extension [Formula: see text].

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21

Dung, Nguyen Viet, and José Luis García. "On the endofiniteness of a key module over pure semisimple rings." Proceedings of the American Mathematical Society 138, no. 07 (July 1, 2010): 2269. http://dx.doi.org/10.1090/s0002-9939-10-10098-7.

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22

Wisbauer, Robert. "On modules with the Kulikov property and pure semisimple modules and rings." Journal of Pure and Applied Algebra 70, no. 3 (March 1991): 315–20. http://dx.doi.org/10.1016/0022-4049(91)90077-f.

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23

Zayed, Maher. "Pure-semisimplicity is preserved under elementary equivalence." Glasgow Mathematical Journal 36, no. 3 (September 1994): 345–46. http://dx.doi.org/10.1017/s0017089500030949.

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Анотація:

In the present note, Σr denotes the class of all right pure semisimple rings (= right pure global dimension zero). It is known that if R ∈ Σr, then R is right artinian and every indecomposable right R-module is finitely generated. The class Σr is not closed under ultraproducts [4]. While Σr is closed under elementary descent (i.e. if S ∈ Σr and R is an elementary subring of S then R ∈ σr) [4], it is an open question whether right pure-semisimplicity is preserved under the passage to ultrapowers [4, Prob. 11.16]. In this note, this question is answered in the affirmative.

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24

Schmidmeier, Markus. "The Local Duality for Homomorphisms and an Application to Pure Semisimple PI-Rings." Colloquium Mathematicum 77, no. 1 (1998): 121–32. http://dx.doi.org/10.4064/cm-77-1-121-132.

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25

ZAYED, MAHER, and AHMED A. ABDEL-AZIZ. "ON MODULES WHICH ARE SUBISOMORPHIC TO THEIR PURE-INJECTIVE ENVELOPES." Journal of Algebra and Its Applications 01, no. 03 (September 2002): 289–94. http://dx.doi.org/10.1142/s0219498802000173.

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Анотація:

In the present paper, modules which are subisomorphic (in the sense of Goldie) to their pure-injective envelopes are studied. These modules will be called almost pure-injective modules. It is shown that every module is isomorphic to a direct summand of an almost pure-injective module. We prove that these modules are ker-injective (in the sense of Birkenmeier) over pure-embeddings. For a coherent ring R, the class of almost pure-injective modules coincides with the class of ker-injective modules if and only if R is regular. Generally, the class of almost pure-injective modules is neither closed under direct sums nor under elementary equivalence. On the other hand, it is closed under direct products and if the ring has pure global dimension less than or equal to one, it is closed under reduced products. Finally, pure-semisimple rings are characterized, in terms of almost pure-injective modules.

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26

Simson*, Daniel. "ON SMALL RIGHT PURE SEMISIMPLE RINGS AND THE STRUCTURE OF THEIR AUSLANDER-REITEN QUIVER." Communications in Algebra 29, no. 7 (May 31, 2001): 2991–3009. http://dx.doi.org/10.1081/agb-5002.

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27

Behboodi, Mahmood, Francois Couchot та Seyed Hossein Shojaee. "Σ-semi-compact rings and modules". Journal of Algebra and Its Applications 13, № 08 (24 червня 2014): 1450069. http://dx.doi.org/10.1142/s0219498814500698.

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Анотація:

In this paper, several characterizations of semi-compact modules are given. Among other results, we study rings whose semi-compact modules are injective. We introduce the property Σ-semi-compact for modules and we characterize the modules satisfying this property. In particular, we show that a ring R is left Σ-semi-compact if and only if R satisfies the ascending (respectively, descending) chain condition on the left (respectively, right) annulets. Moreover, we prove that every flat left R-module is semi-compact if and only if R is left Σ-semi-compact. We also show that a ring R is left Noetherian if and only if every pure projective left R-module is semi-compact. Finally, we consider rings whose flat modules are finitely (singly) projective. For any commutative arithmetical ring R with quotient ring Q, we prove that every flat R-module is semi-compact if and only if every flat R-module is finitely (singly) projective if and only if Q is pure semisimple. A similar result is obtained for reduced commutative rings R with the space Min R compact. We also prove that every (ℵ0, 1)-flat left R-module is singly projective if R is left Σ-semi-compact, and the converse holds if Rℕ is an (ℵ0, 1)-flat left R-module.

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28

Okoh, Frank. "DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS." Communications in Algebra 30, no. 6 (June 19, 2002): 3037–43. http://dx.doi.org/10.1081/agb-120004007.

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