On rings with envelopes and covers regarding to C3, D3 and flat modules

2020 ◽  
pp. 2150228
Author(s):  
Le Van Thuyet ◽  
Phan Dan ◽  
Truong Cong Quynh
Keyword(s):  
Artinian Ring ◽  
Projective Cover ◽  
Perfect Ring ◽  
Flat Modules ◽  
Coherent Ring

In this paper, by taking the class of all [Formula: see text] (or [Formula: see text]) right [Formula: see text]-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes) and a right [Formula: see text]-ring (or a right noetherian [Formula: see text]-ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes). By using isosimple-projective preenvelope, we obtained that if [Formula: see text] is a semiperfect right FGF ring (or left coherent ring), then every isosimple right [Formula: see text]-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.

scholarly journals Acyclic Complexes and Gorenstein Rings

2020 ◽  
Vol 27 (03) ◽  
pp. 575-586
Author(s):  
Sergio Estrada ◽  
Alina Iacob ◽  
Holly Zolt
Keyword(s):  
Analogous Result ◽  
Gorenstein Ring ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Ring ◽  
Acyclic Complexes ◽  
Gorenstein Flat ◽  
Gorenstein Injective

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

scholarly journals A note on existence of envelopes and covers

1996 ◽  
Vol 54 (3) ◽  
pp. 383-390 ◽  
Author(s):  
Jianlong Chen ◽  
Nanqing Ding
Keyword(s):  
Projective Cover ◽  
Direct Products ◽  
Coherent Ring ◽  
Finitely Presented

We prove the following results for a ring R. (a) If C is a class of right R-modules closed under direct summands and isomorphisms, then every right R-module has an epic C-envelope if and only if C is closed under direct products and submodules. (b) If R is left T-coherent and pure injective as a right R-module, then every T-finitely presented right R-module has a T-flat envelope, (c) Let R be a left T-coherent ring and injective right R-modules be T-flat. If every finitely presented left R-module has a flat envelope, then every T-finitely presented right R-module has a projective cover.

scholarly journals A noncommutative generalization of Auslander's last theorem

2005 ◽  
Vol 2005 (9) ◽  
pp. 1473-1480
Author(s):  
Edgar E. Enochs ◽  
Overtoun M. G. Jenda ◽  
J. A. López-Ramos
Keyword(s):  
Projective Cover ◽  
Finitely Generated ◽  
Perfect Ring ◽  
Dualizing Module ◽  
Auslander Class

We show that every finitely generated leftR-module in the Auslander class over ann-perfect ringRhaving a dualizing module and admitting a Matlis dualizing module has a Gorenstein projective cover.

J-COHERENT RINGS

2009 ◽  
Vol 08 (02) ◽  
pp. 139-155 ◽  
Author(s):  
NANQING DING ◽  
YUANLIN LI ◽  
LIXIN MAO
Keyword(s):  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Flat Modules ◽  
Coherent Ring ◽  
Necessary And Sufficient ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied.

Characterizations of Modules

1971 ◽  
Vol 23 (4) ◽  
pp. 608-610
Author(s):  
David J. Fieldhouse
Keyword(s):  
Projective Cover ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Perfect Ring

In this paper we use the Bourbaki [2] conventions for rings and modules. All rings are associative but not necessarily commutative and have a 1; all modules are unital.Bass [1] calls a ring A left perfect if and only if every left A -module has a projective cover, which he shows is equivalent to every flat left A -module being projective. Bass calls a ring A semi-perfect if and only if every finitely generated module has a projective cover and shows that this concept is leftright symmetric.We will define a ring A to be quasi-perfect if and only if every finitely generated flat left A -module is projective.An exercise [6, Exercise 10, p. 136] is given by Lambek to show that every semi-perfect ring is quasi-perfect.

scholarly journals Weakly Projective and Weakly Injective Modules

1994 ◽  
Vol 46 (5) ◽  
pp. 971-981 ◽  
Author(s):  
S. K. Jain ◽  
S. R. López-Permouth ◽  
K. Oshiro ◽  
M. A. Saleh
Keyword(s):  
Positive Integer ◽  
Artinian Ring ◽  
Projective Cover ◽  
Finitely Generated ◽  
Matrix Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Qf Ring ◽  
Finitely Generated Modules

AbstractA module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:(1) R is a left perfect and every weakly projective right R-module is weakly injective.(2) R is a direct sum of matrix rings over local QF-rings.(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.(4) R is right artinian ring and every weakly injective right R-module is weakly projective.(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

Gorenstein n-flat modules and their covers

2014 ◽  
Vol 07 (03) ◽  
pp. 1450051 ◽  
Author(s):  
C. Selvaraj ◽  
R. Udhayakumar ◽  
A. Umamaheswaran
Keyword(s):  
Direct Limit ◽  
Flat Module ◽  
Flat Modules ◽  
Coherent Ring

In this paper, we introduce the notion of Gorenstein n-flat modules and Gorenstein n-absolutely pure modules. First, we prove that the direct limit of Gorenstein n-flat modules over a right n-coherent ring is again a Gorenstein n-flat module. Also we prove that over a right n-coherent ring, any pure submodule of a Gorenstein n-flat module is a Gorenstein n-flat module. Finally, the class of all Gorenstein n-flat left modules over a ring R is a Kaplansky class and then we prove that all left modules over a right n-coherent ring have Gorenstein n-flat covers.

On Gorenstein flat dimension

2015 ◽  
Vol 14 (06) ◽  
pp. 1550096
Author(s):  
Samir Bouchiba
Keyword(s):  
Global Dimension ◽  
Arbitrary Ring ◽  
Cohomological Invariants ◽  
Flat Dimension ◽  
Flat Modules ◽  
Coherent Ring ◽  
Left And Right ◽  
Gorenstein Flat ◽  
Projectively Resolving

The theory of Gorenstein flat dimension is not complete since it is not yet known whether the category 𝒢ℱ(R) of Gorenstein flat modules over a ring R is projectively resolving or not. Besides, it arises from recent investigations on this subject that there exists several ways of measuring the Gorenstein flat dimension of modules which turn out to coincide with the usual one in the case where 𝒢ℱ(R) is projectively resolving. These alternate procedures yield new invariants which enjoy very nice behavior for an arbitrary ring R. In this paper, we introduce and study one of these invariants called the cover Gorenstein flat dimension of a module M and denoted by CGfd R(M). This new entity stems from a sort of a Gorenstein flat precover of M. First, for each R-module M, we prove that Gfd R(M) ≤ CGfd R(M) for each R-module M with [Formula: see text] whenever CGfd R(M) is finite. Also, we show that 𝒢ℱ(R) is projectively resolving if and only if the Gorenstein flat dimension and the introduced cover Gorenstein flat dimension coincide. In particular, if R is a right coherent ring, then CGfd R(M) = Gfd R(M) for any R-module M. As a consequence, we prove that if R is a left and right GF-closed, then the Gorenstein weak global dimension of R is left–right symmetric and it is related to the cohomological invariants leftsfli(R) and rightsfli(R) by the formula [Formula: see text]

Coherent Ring Currents in Chiral Aromatic Molecules Induced by Linearly Polarized UV Laser Pulses

2013 ◽  
Vol 543 ◽  
pp. 381-384 ◽  
Author(s):  
Manabu Kanno ◽  
Hirohiko Koho ◽  
Hirobumi Mineo ◽  
Sheng Hsien Lin ◽  
Yuichi Fujimura
Keyword(s):  
Aromatic Ring ◽  
Quantum Dynamics ◽  
Ring Current ◽  
Molecular System ◽  
Laser Pulses ◽  
Uv Laser ◽  
Aromatic Molecules ◽  
Ring Currents ◽  
Linearly Polarized ◽  
Coherent Ring

In recent years, laser control of electrons in molecular system and condensed matter has attracted considerable attention with rapid progress in laser science and technology [. In particular, control of π-electron rotation in photo-induced chiral aromatic molecules has potential utility to the next-generation ultrafast switching devices. In this paper, we present a fundamental principle of generation of ultrafast coherent ring currents and the control in photo-induced aromatic molecules. This is based on quantum dynamics simulations of π-electron rotations and preparation of unidirectional angular momentum by ultrashort UV laser pulses properly designed. For this purpose, we adopt 2,5-dichloro [(3,6) pyrazinophane (DCPH) fixed on a surface, which is a real chiral aromatic molecule with plane chirality. Here π electrons can be rotated along the aromatic ring clockwise or counterclockwise by irradiation of a linearly polarized laser pulse with the properly designed photon polarization direction and the coherent ring current with the definite direction along the aromatic ring is prepared. This is contrast to ordinary ring current in an achiral aromatic ring molecule with degenerate electronic excited state, which is prepared by a circularly polarized laser [2]. In this case, π electrons rotate along the Z-axis of the laboratory coordinates, while for the present case electrons rotate along the z-axis in molecular Cartesian coordinates. It should be noted that signals originated from the coherent ring currents prepared by linearly polarized ultrashort UV lasers are specific to the chiral molecule of interest.

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