ALL GORENSTEIN HEREDITARY RINGS ARE COHERENT

2014 ◽  
Vol 13 (04) ◽  
pp. 1350140 ◽  
Author(s):  
ZENGHUI GAO ◽  
FANGGUI WANG
Keyword(s):  
Projective Module ◽  
Finitely Generated ◽  
Hereditary Ring

A ring R is called Gorenstein hereditary (G-hereditary) if every submodule of a projective module is Gorenstein projective (i.e. Ggldim (R) ≤ 1). In this paper, we settle a question raised by Mahdou and Tamekkante in [On (strongly) Gorenstein (semi)hereditary rings, Arab. J. Sci. Eng.36 (2011) 436] about the coherence of G-hereditary rings. It is shown that a ring R is Gorenstein semihereditary if and only if every finitely generated submodule of a projective module is Gorenstein projective. As a consequence of this result, we have that every G-hereditary ring R is coherent.

Semiregular Modules and Rings

1976 ◽  
Vol 28 (5) ◽  
pp. 1105-1120 ◽  
Author(s):  
W. K. Nicholson
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Projective Module ◽  
Structure Theorem ◽  
Projective Cover ◽  
Finitely Generated

Mares [9] has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. More recently Zelmanowitz [16] has called a module regular if every finitely generated submodule is a projective direct summand. In the present paper a class of semiregular modules is introduced which contains all regular and all semiperfect modules. Several characterizations of these modules are given and a structure theorem is proved. In addition several theorems about regular and semiperfect modules are extended.

scholarly journals When every projective module is a direct sum of finitely generated modules

2007 ◽  
Vol 315 (1) ◽  
pp. 454-481 ◽  
Author(s):  
Warren Wm. McGovern ◽  
Gena Puninski ◽  
Philipp Rothmaler
Keyword(s):  
Projective Module ◽  
Finitely Generated ◽  
Direct Sum ◽  
Finitely Generated Modules

scholarly journals Polynomial rings and their projective modules

1989 ◽  
Vol 113 ◽  
pp. 121-128 ◽  
Author(s):  
Dorin Popescu
Keyword(s):  
Noetherian Ring ◽  
Projective Module ◽  
Polynomial Rings ◽  
Projective Modules ◽  
Central Question ◽  
Finitely Generated

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.

Strongly D2 modules and their applications

2021 ◽  
pp. 2250071
Author(s):  
Mingzhao Chen ◽  
Hwankoo Kim ◽  
Fanggui Wang
Keyword(s):  
Positive Integer ◽  
Principal Ideal ◽  
Projective Module ◽  
Negative Answer ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Finitely Generated Modules ◽  
Structural Characterizations ◽  
Semihereditary Rings

An [Formula: see text]-module [Formula: see text] is called strongly [Formula: see text] if [Formula: see text] is a [Formula: see text] (equivalently, direct projective) module for every positive integer [Formula: see text]. In this paper, we consider the class of quasi-projective [Formula: see text]-modules, the class of strongly [Formula: see text] [Formula: see text]-modules and the class of [Formula: see text]-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations of strongly [Formula: see text] modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly [Formula: see text] modules.

Pure Projective Approximations

2009 ◽  
Vol 146 (1) ◽  
pp. 83-94 ◽  
Author(s):  
IVO HERZOG ◽  
PHILIPP ROTHMALER
Keyword(s):  
Finite Type ◽  
Projective Module ◽  
Torsion Class ◽  
Finitely Generated ◽  
Bijective Correspondence ◽  
Functor Category ◽  
Finitely Presented ◽  
Covariantly Finite Subcategory

AbstractA notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

Σ-Rickart modules

2019 ◽  
Vol 19 (11) ◽  
pp. 2050207
Author(s):  
Gangyong Lee ◽  
Mauricio Medina-Bárcenas
Keyword(s):  
Direct Summand ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Hereditary Ring ◽  
Direct Sum ◽  
Factor Module ◽  
Rickart Modules

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50’s. In this paper, we study the notion of a [Formula: see text]-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module [Formula: see text] is called [Formula: see text]-Rickart if every direct sum of copies of [Formula: see text] is Rickart. It is shown that any direct summand and any direct sum of copies of a [Formula: see text]-Rickart module are [Formula: see text]-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring [Formula: see text] is right hereditary if and only if every submodule of any projective right [Formula: see text]-module is projective if and only if every factor module of any injective right [Formula: see text]-module is injective. Also, we have a characterization of a finitely generated [Formula: see text]-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.

scholarly journals Decompositions of modules into projective modules and CS-modules

2000 ◽  
Vol 62 (1) ◽  
pp. 159-164
Author(s):  
Somyot Plubtieng
Keyword(s):  
Projective Module ◽  
Injective Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Module ◽  
Direct Sum ◽  
Continuous Module

Let M be a right R-module. It is shown that M is a locally Noetherian module if every finitely generated module in σ[M] is a direct sum of a projective module and a CS-module. Moreover, if every module in σ[M] is a direct sum of a projective module and a CS-module, then every module in σ[M] is a direct sum of modules which are either indecomposable projective or uniform Σ-quasi-injective. In particular, if every module in σ[M] is a direct sum of a projective module and a quasi-continuous module, then every module in σ[M] is a direct sum of a projective module and a quasi-injective module.

scholarly journals On Trace for Modules

1970 ◽  
Vol 40 ◽  
pp. 121-131 ◽  
Author(s):  
Howard B. Beckwith
Keyword(s):  
Characteristic Zero ◽  
Projective Module ◽  
Canonical Homomorphism ◽  
Projective Modules ◽  
Finitely Generated ◽  
Group Characters ◽  
Classical Mathematics ◽  
Square Matrix

Classically, trace was defined as the sum of the diagonal entries of a square matrix with entries in a field. This notion played an important role in classical mathematics, e.g. in the theory of algebras over a field of characteristic zero, and in the theory of group characters (as in [1]). A generalization to endomorphisms of a finitely generated projective module over any ring R with unit is well-known. For such a module P the canonical homomorphism Ψ: P* ⊗ P→ EndR(P) is an isomorphism. Then the composite ε˚Ψ-1: EndR(P) → P* ↗ P→ R, where e denotes “evaluation”, is a homomorphism which coincides with the classical trace whenever P is free. This version of trace has been used by Hattori [3] and others to study projective modules. However, this approach to trace is limited to the finitely generated projective modules, since it can be shown that Ψ is an isomorphism if and only if P is finitely generated and projective.

scholarly journals QF-3 endomorphism rings of Σ-quasi-projective modules

1988 ◽  
Vol 30 (2) ◽  
pp. 215-220 ◽  
Author(s):  
José L. Gómez Pardo ◽  
Nieves Rodríguez González
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Sufficient Conditions ◽  
Torsion Theory ◽  
Projective Modules ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Quotient Module ◽  
Qf Ring ◽  
Necessary And Sufficient

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

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