scholarly journals Projective summands in generators

1982 ◽  
Vol 86 ◽  
pp. 203-209 ◽  
Author(s):  
David Eisenbud ◽  
Wolmer Vasconcelos ◽  
Roger Wiegand
Keyword(s):  
Homomorphic Image ◽  
Polynomial Ring ◽  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Positive Direction ◽  
Chain Conditions ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Category Of Modules

An R-module M is a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M(k) has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free summands locally. Of course, generators can fail to have free summands (e.g., over Dedekind domains), and we ask whether they necessarily have non-zero projective summands. The answer is “yes” for rings of dimension 1, as we point out in § 3, and for the polynomial ring in one variable over a Dedekind domain. In § 1 we show that for 2-dimensional rings the answer is intimately connected with the structure of projective modules. Our main result in the positive direction, Theorem 1.3, grew out of the attempt, in conversations with T. Stafford, to understand the case R = k[x, y]. In § 2 we give examples of rings having generators with no projective summands. The last section contains miscellaneous observations, some of them on rings without chain conditions.

scholarly journals Half-exact coherent functors over Dedekind domains

2019 ◽  
Vol 18 (05) ◽  
pp. 1950099
Author(s):  
Adson Banda
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Dedekind Domain ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Coherent Functor ◽  
Coherent Functors

Let [Formula: see text] be a principal ideal domain (PID) or more generally a Dedekind domain and let [Formula: see text] be a coherent functor from the category of finitely generated [Formula: see text]-modules to itself. We classify the half-exact coherent functors [Formula: see text]. In particular, we show that if [Formula: see text] is a half-exact coherent functor over a Dedekind domain [Formula: see text], then [Formula: see text] is a direct sum of functors of the form [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is a finitely generated projective [Formula: see text]-module, [Formula: see text] a nonzero prime ideal in [Formula: see text] and [Formula: see text].

Quasi-strongly Gorenstein projective modules

2019 ◽  
Vol 18 (10) ◽  
pp. 1950182
Author(s):  
Kui Hu ◽  
Fanggui Wang ◽  
Longyu Xu ◽  
Dechuan Zhou
Keyword(s):  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Dedekind Domains ◽  
Gorenstein Projective Modules ◽  
Semihereditary Rings

In this paper, we introduce the class of quasi-strongly Gorenstein projective modules which is a particular subclass of the class of finitely generated Gorenstein projective modules. We also introduce and characterize quasi-strongly Gorenstein semihereditary rings. We call a quasi-strongly Gorenstein semihereditary domain a quasi-SG-Prüfer domain. A Noetherian quasi-SG-Prüfer domain is called a quasi-strongly Gorenstein Dedekind domain. Let [Formula: see text] be a field and [Formula: see text] be an indeterminate over [Formula: see text]. We prove that every ideal of the ring [Formula: see text] is strongly Gorenstein projective. We also show that every ideal of the ring [Formula: see text] (respectively, [Formula: see text]) is strongly Gorenstein projective. These domains are examples of quasi-strongly Gorenstein Dedekind domains.

Prime uniserial modules and rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
Author(s):  
M. Behboodi ◽  
Z. Fazelpour
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Prime Submodules ◽  
Uniserial Modules ◽  
Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

Arithmetic modules over generalized Dedekind domains

2020 ◽  
pp. 2250045
Author(s):  
Indah Emilia Wijayanti ◽  
Hidetoshi Marubayashi ◽  
Iwan Ernanto ◽  
Sutopo
Keyword(s):  
Free Module ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Dedekind Domains ◽  
Torsion Free Module ◽  
Polynomial Module ◽  
Torsion Free

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

scholarly journals Almost triangular matrices over Dedekind domains

1999 ◽  
Vol 22 (4) ◽  
pp. 761-764
Author(s):  
Frank Demeyer ◽  
Haniya Kakakhail
Keyword(s):  
Dedekind Domain ◽  
Triangular Matrices ◽  
Direct Sum ◽  
Dedekind Domains

Every matrix over a Dedekind domain is equivalent to a direct sum of matricesA=(ai,j), whereai,j=0wheneverj>i+1.

scholarly journals Nice Bases of QTAG-Modules

2021 ◽  
pp. 1-5
Author(s):  
Fahad Sikander ◽  
Tanveer Fatima ◽  
Ayazul Hasan
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Divisible Hull

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).

The Cancellation Property of Projective Modules

2006 ◽  
Vol 13 (04) ◽  
pp. 617-622 ◽  
Author(s):  
Hongbo Zhang ◽  
Wenting Tong
Keyword(s):  
Projective Modules ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Cancellation Property ◽  
Dedekind Domains

An R-module M is said to have the cancellation property provided that M⊕ B ≅ M⊕ C implies B ≅ C for any pair of R-modules B and C. We obtain a characterization of the cancellation property for projective R-modules. With this result, it is proved that Dedekind domains have the cancellation property; and if R is a Prüfer domain, then R⊕ B ≅ R⊕ C implies B ≅ C for any pair of finitely generated R-modules B and C.

scholarly journals On Quasi-h-Dense Submodules and h-Pure Envelopes of QTAG Modules

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Alveera Mehdi ◽  
Fahad Sikander ◽  
Firdhousi Begum
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Pure Submodules

A module M over an associative ring R with unity is a QTAG module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which h-pure submodules and high submodules are significant. A submodule N is quasi-h-dense in M if M/K is h-divisible, for every h-pure submodule K of M, containing N. Here we study these submodules and obtain some interesting results. Motivated by h-neat envelope, we also define h-pure envelope of a submodule N as the h-pure submodule K⊇N if K has no direct summand containing N. We find that h-pure envelopes of N have isomorphic basic submodules, and if M is the direct sum of uniserial modules, then all h-pure envelopes of N are isomorphic.

Structures of QTAG-modules determined by their submodules

2021 ◽  
Vol 65 (3) ◽  
pp. 38-45
Author(s):  
Ayazul Hasan
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper, we study the existence of several classes C of QTAG-modules which satisfy the property that M belongs to C uniquely when M/N belongs to C provided that N is a finitely generated submodule of the QTAG-module.

D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS

2020 ◽  
pp. 1-27
Author(s):  
GABRIELLA D′ESTE ◽  
DERYA KESKİN TÜTÜNCÜ ◽  
RACHID TRIBAK
Keyword(s):  
Direct Summand ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Dedekind Domain ◽  
Projective Modules ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Rickart Modules

Abstract A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.

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