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On almost excellent extensions
Abstract
Assuming S ≥ R is an almost excellent extension we prove that (1) if MS is a right S-module then the equalities pd(MS) = pd(MR), id(MS) = id(MR) and fd(MS) = fd(MR) hold; and (2) if either ring is right FS, right hereditary, right (left) semihereditary, right SI, right GV, right S3I, or right almost artinian (noetherian), then so is the other. We also consider dual Goldie dimensions over a finite normalizing extension.
... A natural next step is to consider generalisations of excellent extensions in Section 6. In particular we focus on finite normalising extensions and almost excellent extensions defined by Xue [Xue96]. ...
... However there are generalisations of excellent extensions which are not. We focus on two of these, namely finite normalising extensions and almost excellent extensions [Xue96]. ...
... Almost excellent extensions were defined and studied by Xue [Xue96]. ...
Given the unbounded derived module category of a ring $A$, we consider the triangulated subcategory closed under arbitrary coproducts generated by injective modules. Similarly we also look at the triangulated subcategory closed under arbitrary products cogenerated by projective modules. For a ring construction $f(A)$, we ask whether $A$ being generated by its injective modules implies $f(A)$ is also generated by its injective modules, and vice versa. Similarly we ask the question with projective modules and cogeneration. In this paper we show when these statements are true for ring constructions including recollements, Frobenius extensions and module category equivalences.
... Many authors have studied the invariant properties of rings under excellent extensions (see e.g. Bonami, 1984;Feng, 1997;Passman, 1977;Xue, 1996;Xiao, 1994). It has been known that many important homological properties, such as the (weak) global dimension of rings, the projectivity, injectivity and flatness of modules and so on, are invariant under excellent extensions (Xue, 1996). ...
... Bonami, 1984;Feng, 1997;Passman, 1977;Xue, 1996;Xiao, 1994). It has been known that many important homological properties, such as the (weak) global dimension of rings, the projectivity, injectivity and flatness of modules and so on, are invariant under excellent extensions (Xue, 1996). ...
... (1) (Xue, 1996 (2) (Xue, 1996, Lemma 1.1) Let N be a R-module, then N R |(N ⊗ R S ) R ; ...
... A ring S is said to be an almost excellent extension of a ring R [14,15] if the following conditions are satisfied: ...
... The concept of excellent extension was introduced by Passman [7] and named by Bonami [2]. The notion of almost excellent extensions was introduced and studied in [14,15] as a non-trivial generalization of excellent extensions. ...
... Note that M S (resp., N S ) is isomorphic to a direct summand of Hom R (S, M ) (resp., Hom R (S, N )) by [15,Lemma 1.1 (2)], and so we have the following exact commutative diagram ...
In this paper, we study the F P -projective dimension under changes of rings, es-pecially under (almost) excellent extensions of rings. Some descriptions of F P -injective envelopes are also given.
... S is called a ÿnite normalizing extension of R if there exist elements a 1 ; a 2 ; : : : ; a n ∈ S such that S = n i=1 a i R where a i R=Ra i for i=1; 2; : : : ; n. Finite normalizing extensions have been studied in many papers such as [12,22,14,17,18,20,21]. S is called a free normalizing extension of R if S = n i=1 a i R is a normalizing extension of R and S is free with basis {a 1 = 1; a 2 ; : : : ; a n } as both a right R-module and a left R-module. ...
... Examples include ÿnite matrix rings and the crossed product R * G, where G is a ÿnite group with |G| −1 ∈ R. S is said to be an almost excellent extension of R if S is a ÿnite normalizing extension such that S R and R S are projective as a right R-module and a left R-module, respectively, while S is right R-projective. Almost excellent extensions were introduced and studied by Xue [20,21] as a non-trivial generalization of excellent extensions. Let R = (T R ; F R ) and S = (T S ; F S ) be torsion theories for right R-modules and right S-modules, respectively, and let ' : R → S be a ring extension. ...
... In this section, we shall consider some special rings relative to ÿnite normalizing extensions (excellent extensions and almost excellent extensions) of rings and TH (THT )-extensions of torsion theories and obtain some applications of TH -extensions. In addition, we answer Xue's [20] open problems in the positive. Xue [20] proved that if S ¿ R is an almost excellent extension, then S is right coherent if and only if R is right coherent. ...
Let S be a ring extension of R, and let τR=(TR,FR) and τS=(TS,FS) be torsion theories for right R-modules and right S-modules respectively. Using functors HomR(−,S) and −⊗RS, we get the connection between τR and τS, introduce torsion theory extensions related to ring extensions and establish some relations between finite normalizing extensions and torsion theory extensions. Furthermore, we consider some applications of torsion theory extensions and generalize some results of excellent extensions (almost excellent extensions) to the setting of torsion theories.
... Such extensions of rings are vital since they include two important classes of extensions of rings, that is, finite matrix rings and skew group rings Λ * G where the finite group G satisfies the condition |G| −1 ∈ Λ (see Example 2.2 below for the details). Many authors have studied the invariant properties of rings under excellent extensions [8,15,23,25,26,29,33]. It has been known that many important homological properties, such as the (weak) global dimension of rings, the projectivity, injectivity and flatness of modules and so on, are invariant under excellent extensions [23,29]. ...
... Many authors have studied the invariant properties of rings under excellent extensions [8,15,23,25,26,29,33]. It has been known that many important homological properties, such as the (weak) global dimension of rings, the projectivity, injectivity and flatness of modules and so on, are invariant under excellent extensions [23,29]. ...
... In addition, compare the definition of the weak excellent extension with that of the almost excellent extension in [29]. ...
In this paper, we introduce the notion of weak excellent extensions of rings as a generalization of that of excellent extensions of rings. Let Gamma be a weak excellent extension of an Artinian algebra Lambda. We prove that if Lambda is of finite representation type (resp. CM-finite, CM-free), then so is Gamma; furthermore, if Gamma is an excellent extension of Lambda. then the converse also holds true. We also study when the representation dimension of an Artinian algebra is invariant under excellent extensions. C) 2012 Elsevier Inc. All rights reserved.
... If M R is finitely generated torsionless, by Lemma 3.1, then (M ⊗ R S) S is finitely generated torsionless. In view of Lemma 1.1 of [17], M S is isomorphic to a direct summand of (M ⊗ R S) S , so M S is finitely generated torsionless. ...
... (1) The proof is similar to that of Corollary 1.5 of [17]. ...
... Note that (K ⊗ R S) S is finitely generated and (F ⊗ R S) S is finitely generated free, then (M ⊗ R S) S is finitely presented. But M S is isomorphic to a direct summand of (M ⊗ R S) S by Lemma 1.1 of [17], so M S is finitely presented. 2 ...
We study, in this article, the FGT-injective dimensions of Π-coherent rings. If R is right Π-coherent, and TI (resp. TF) stands for the class of FGT-injective (resp. FGT-flat) R-modules (n ≥ 0), we show that the following are equivalent:
(1)
FGT - Id
R
(R) ≤ n
(2)
If 0 → M → F
0 → F
1 → ... is a right TF-resolution of left R-module M, then the sequence is exact at F
k
for k ≥ n − 1
(3)
For every flat right R-module F, there is an exact sequence 0 → F → A
0 → A
1 → ... → A
n
→ 0 with each A
i
∈ TI
(4)
For every injective left R-module A, there is an exact sequence 0 → F
n
→ ... → F
1 → F
0 → A → 0 with each F
i
∈ TF
(5)
If ... → I
1 → I
0 → M → 0 is a minimal left TI-resolution of a right R-module M, then the sequence is exact at I
k
for k ≥ n − 1.
Further, we characterize such homological dimension in terms of TI-syzygy and TF-cosyzygy of modules. Finally, we consider almost excellent extensions of rings. These extend the corresponding results in
[10] as well.
KeywordsΠ-coherent ring-FGT-injective dimension-resolution-almost excellent extension
... In this section we provide sufficient conditions on a ring extension f : B → A such that if injectives generate for A, then injectives generate for B, and vice versa. In particular, we focus on Frobenius extensions [21,25], almost excellent extensions [40] and trivial extensions. In Section 5.1 we apply the results in this section to the arrow removal operation defined in [14,Section 4]. ...
... Almost excellent extensions were defined by Xue [40] as a generalisation of excellent extensions, which were first introduced by Passman [26]. The interaction of excellent extensions with various properties of rings has been studied in [17]. ...
We consider the smallest triangulated subcategory of the unbounded derived module category of a ring that contains the injective modules and is closed under set indexed coproducts. If this subcategory is the entire derived category, then we say that injectives generate for the ring. In particular, we ask whether, if injectives generate for a collection of rings, do injectives generate for related ring constructions, and vice versa. We provide sufficient conditions for this statement to hold for various constructions including recollements, ring extensions and module category equivalences.
... The notion of almost excellent extension was introduced and studied in [6] as a non-trivial generalization of excellent extensions. ...
... In the following, we mainly consider the properties of F P -projective modules and F P -injective modules under an almost excellent extensions of rings. Lemma 2.1 [6] Let S be an almost excellent extension of R and M S a right R-module. Then [4] Let {M i } be a family of left R-modules. ...
Let R be a ring and S an almost excellent extension of R. In this pa-per, we discussed the homological properties of F P -projective modules and F P -injective modules under the extension of rings, which mainly including almost excellent extensions, Polynomial extensions, Morita equivalences and localizations.
... A ring S is said to be an almost excellent extension of a ring R [16,17] if the following conditions are satisfied: ...
... The concept of excellent extension was introduced by Passman [21] and named by Bonami [22]. The notion of almost excellent extension was introduced and studied in [16,17] as a non-trivial generalization of excellent extension. Examples include n × n matrix rings [21], and crossed products R * G where G is a finite group with |G| −1 ∈ R (see [23]). ...
This article is concerned with the strongly Gorenstein flat dimensions of modules and rings. We show that the dimension has nice properties when the ring is coherent, and extend the well-known Hilbert’s syzygy theorem to the strongly Gorenstein flat dimensions of rings. Also, we investigate the strongly Gorenstein flat dimensions of direct products of rings and (almost) excellent extensions of rings.
... Let C S and C R be classes of right Smodules and right R-modules. In Section 2 we use functors Hom R (S, −) and − ⊗ R S to get the connection between C R and C S , and introduce class extensions related to ring extensions, that is, C S is a T-extension, an H-extension of C R. For an almost excellent extension S ≥ R, if C S is a T-extension or H-extension of C R , we show that C S is a (resp., monomorphic, epimorphic, special) preenveloping class if and only if so is C R. As corollaries, some results by Xue [6,12] can be obtained. In Section 3, let C S = (A S , B S ) and C R = (A R , B R ) be cotorsion pair of right S-modules and right R-modules respectively. ...
... Let A S (resp., B S )= Mod-S, and A R (resp., B R )= Mod-R. We have the following corollaries obtained by Xue in [6] and [16]. ...
Assume that S is an almost excellent extension of R. Using functors HomR(S,-) and -⊗RS, we establish some connections between classes of modules CR and CS, cotorsion pairs (AR, BR) and (AR, BR). If CS is a T-extension or (and) H-extension of CR, we show that CS is a (resp., monomorphic, epimorphic, special) preenveloping class if and only if so is CR. If (AS, BS) is a TH-extension of (AR, BR), we obtain that (AS, BS) is complete (resp., of finite type, of cofinite type, hereditary, perfect, n-tilting) if and only if so is (AR, BR). © 2009 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer Berlin Heidelberg.
Let $R$ be a ring graded by a group $G$ and $n\geq1$ an integer. We introduce the notion of $n$-FCP-gr-projective $R$-modules and by using of special finitely copresented graded modules, we investigate that (1) there exist some equivalent characterizations of $n$-FCP-gr-projective modules and graded right modules of $n$-FCP-gr-projective dimension at most $k$ over $n$-gr-cocoherent rings, (2) $R$ is right $n$-gr-cocoherent if and only if for every short exact sequence $0 \rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ of graded right $R$-modules, where $B$ and $C$ are $n$-FCP-gr-projective, it follows that $A$ is $n$-FCP-gr-projective if and only if ($gr$-$\mathcal{FCP}_{n}$, $gr$-$\mathcal{FCP}_{n}^{\bot}$) is a hereditary cotorsion theory, where $gr$-$\mathcal{FCP}_n$ denotes the class of $n$-FCP-gr-projective right modules. Then we examine some of the conditions equivalent to that each right $R$-module is $n$-FCP-gr-projective.
Several authors have introduced various type of coherent-like rings and proved analogous results on these rings. It appears that all these relative coherent rings and all the used techniques can be unified. In [2], several coherent-like rings are unified. In this manuscript we continue this work and we introduce coherent-like module which also emphasizes our point of view by unifying the existed relative coherent concepts. Several classical results are generalized and some new results are given.
We prove that the weak equivalences, cofibrations and fibrations in Gillespie’s flat model structure are invariant under excellent extensions of rings. As applications, the flat and cotorsion dimensions of unbounded complexes along excellent extensions are investigated, and moreover, some homological invariant properties under excellent extensions are studied in the present framework of complexes.
Let \(\mathcal {A}\) be an abelian category having enough projective objects and enough injective objects. We prove that if \(\mathcal {A}\) admits an additive generating object, then the extension dimension and the weak resolution dimension of \(\mathcal {A}\) are identical, and they are at most the representation dimension of \(\mathcal {A}\) minus two. By using it, for a right Morita ring Λ, we establish the relation between the extension dimension of the category mod Λ of finitely generated right Λ-modules and the representation dimension as well as the right global dimension of Λ. In particular, we give an upper bound for the extension dimension of mod Λ in terms of the projective dimension of certain class of simple right Λ-modules and the radical layer length of Λ. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.
Let R be a ring. A left R-module M is said to be sfp-injective if, for every exact sequence 0 → K → L with K and L super finitely presented left R-modules, the induced sequence Hom(L, M) → Hom(K, M) → 0 is exact. A right R-module N is called sfp-flat if, for every exact sequence 0 → K → L with K and L super finitely presented left R-modules, the induced sequence 0 → N ⊗ K → N ⊗ L is exact. We study precovers and preenvelopes by sfp-injective and sfp-flat modules, including their properties under (almost) excellent extensions of rings.
A morphism f: M → N of left R-modules is called a phantom morphism if the induced morphism for every (finitely presented) right R-module A. Similarly, a morphism g: M → N of left R-modules is said to be an Ext-phantom morphism if the induced morphism for every finitely presented left R-module B. We prove that every left R-module has a phantom preenvelope if R is a right coherent ring and every left R-module has an Ext-phantom cover if R is a left coherent ring. In addition, we investigate the properties of precovers and preenvelopes by phantom and Ext-phantom morphisms under change of rings.
In this paper, we introduce the notion of fp n -injective and fp n -flat modules, and investigate their properties. In particular, we prove that every left R -module has an fp n -injective cover (resp. preenvelope) and every right R -module has an fp n -flat cover (resp. preenvelope). We also study the exchange properties of fp n -injective and fp n -flat modules, as well as fp n -injective precovers (resp. preenvelopes) and fp n -flat precovers (resp. preenvelopes), under (almost) excellent extensions of rings.
In this paper, we consider G C-projective, injective and flat modules and dimensions under change of rings.
In this paper, we study the right Gorenstein global dimension and the r.IFD(–) of an (almost) excellent extension of ring.
In this paper, we mainly investigate the global cotorsion dimension under the excellent extension of rings.
We show that if $S$ is an excellent extension of $R$, then $\mathrm{cot.D}(S)=\mathrm{cot.D}(R)$.
Furthermore, some known results, such as Corollary 3.8 and 3.12, can be also obtained as direct corollaries of our theorem.
In this paper the n --injective (pre)envelopes and the n --projective (pre)covers are investigated, which generalize and include some known envelopes and covers. Various equivalent characterizations and sufficient conditions of their existence are provided. In addition, the (special) -injective (pre)envelopes are discussed under almost (or quasi)excellent extensions of rings, some new or extending results on the -envelopes and the -covers are obtained.
We introduce the concepts of (n, d)-injective and (n, d)-flat as generalizations of injective, flat modules and homological dimensions, and use them to characterize right n-coherent rings and right (weak) (n, d)-rings in various way. Some known results can be obtained as corollaries.
Let ℒ be a class of right R-modules. The notions of ℒ-injectivity and ℒ-flatness are used to investigate right ℒ-Noetherian rings, right ℒ-hereditary rings, and right ℒ-coherent rings. For an almost excellent extension S ≥ R, if either ring is the above-mentioned ring, so is the other. Specializing the class ℒ, some known results can be obtained and extended as corollaries.
Let , be classes of modules, we introduce and characterize orthogonal dimensions of modules and rings by the orthogonal classes of modules , , and T . For an almost excellent extension S ≥ R, we give the criteria to test some identities of orthogonal dimensions of modules and rings. As corollaries, some known results are obtained or extended.
A ringR is left co-semihereditary (strongly left co-semihereditary) if every finitely cogenerated factor of a finitely cogenerated
(arbitrary) injective leftR-module is injective. A left co-semihereditary ring, which is not strongly left co-semihereditary, is given to answer a question
of Miller and Tumidge in the negative. If
R
U
S
defines a Morita duality,R is proved to be left co-semihereditary (left semihereditmy) if and only ifS is right semihereditary (right co-semihereditary). Assuming thatS⩾R is an almost excellent extension,S is shown to be (strongly) right co-semihereditary if and only ifR is (strongly) right co-semihereditary.
A left R-module M is said to be f p-injective if, for every monomorphism K → L with K and L finitely presented left R-modules, Hom(L, M) → Hom(K, M) is an epimorphism. A right R-module N is called f p-flat if, for every monomorphism K → L with K and L finitely presented left R-modules, NÄK® NÄL{N\otimes K\rightarrow N\otimes L} is a monomorphism. In this note, we study precovers and preenvelopes by f p-injective and f p-flat modules, including their properties under (almost) excellent extensions of rings. In addition, we also introduce and
investigate f p-projective modules.
Keywords
f p-Injective module–
f p-Flat module–
f p-Projective module–Precover–Preenvelope–(almost) Excellent extension
We define the presented dimensions for modules and rings to measure how far away
a module is from having an infinite finite presentation and develop ways to compute
the projective dimension of a module with a finite presented dimension and the right
global dimension of a ring. We also make a comparison of the right global dimension,
the weak global dimension, and the presented dimension and divide rings into four
classes according to these dimensions.
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