On Presented Dimensions of Modules and Rings

Abstract

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.

1. Introduction

Let 𝑅 be a ring and 𝑛 a nonnegative integer. Following [1, 2], a right 𝑅-module 𝑀 is called 𝑛-presented in case it has a finite 𝑛-presentation, that is, there is an exact sequence of right 𝑅-modules𝐹_𝑛→𝐹_𝑛−1→⋯→𝐹₁→𝐹₀→𝑀→0, (1.1) where each 𝐹_𝑖 is a finitely generated free, equivalently projective, right 𝑅-module. A module is 0-presented (resp., 1-presented) if and only if it is finitely generated (resp., finitely presented), and each 𝑚-presented module is 𝑛-presented for 𝑚≥𝑛. A ring 𝑅 is called right 𝑛-coherent in case every 𝑛-presented right 𝑅-module is (𝑛+1)-presented. It is easy to see that 𝑅 is right 0-coherent (resp., 1-coherent) if and only if it is right Noetherian (resp., coherent), and every 𝑛-coherent ring is 𝑚-coherent for 𝑚≥𝑛.

As in [1, 3], we set 𝜆_𝑅(𝑀)=sup{𝑛∣𝑀 has a finite 𝑛-presentation} and note that 𝜆_𝑅(𝑀)≥𝑛 is a way to express how far away a module 𝑀 is from having an infinite finite presentation. Clearly every finitely generated projective module 𝑀 has an infinite finite presentation, that is, 𝜆_𝑅(𝑀)=∞. The lambda dimension of a ring 𝑅 is the infimum of the set of integers 𝑛 such that every 𝑅-module having a finite 𝑛-presentation has an infinite finite presentation. It was studied extensively by Vasconcelos in [3], where it was denoted by 𝜆-dim(𝑅). Note that 𝑅 is right 𝑛-coherent if and only if 𝜆-dim(𝑅)≤𝑛 and if and only if every 𝑛-presented module has an infinite finite presentation.

Ng [4] defined the finitely presented dimension of a module 𝑀 as f.p.dim(𝑀) = inf{𝑛∣ there exists an exact sequence 𝑃_𝑛+1→𝑃_𝑛→⋯→𝑃₀→𝑀→0 of 𝑅-modules, where each 𝑃_𝑖 is projective, and 𝑃_𝑛+1,𝑃_𝑛 are finitely generated}, which measures how far away a module is from being finitely presented. Motivated by this, we define a dimension, called presented dimension, for modules and rings in this paper. It measures how far away a module is from having an infinite finite presentation and how far away a ring is from being Noetherian. In Section 2, we give the definitions and show the properties of presented dimensions. In Section 3, using strongly presented modules, we give the structure of modules with presented dimensions ≤1 and develop ways to compute the projective dimension of a module with a finite presented dimension and the right global dimension of a ring. In Section 4, we define the presented dimension of a ring, 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. In Section 5, we provide the properties of presented dimensions of modules and rings under an almost excellent extension of rings.

Throughout rings are associative with identity, modules are unitary right 𝑅-modules, and homomorphisms are module homomorphisms. The notations pd(𝑀),id(𝑀), andfd(𝑀)denote the projective, injective, flat dimension of 𝑀, and rgD(𝑅), wD(𝑅) denote the right global dimension, weak global dimension, respectively. For other definitions and notations in this paper we refer to [5, 6].

2. Presented Dimensions of Modules

Definition 2.1. Let 𝑀 be a right 𝑅-module, define the presented dimension of 𝑀 as follows: FPd(𝑀)=inf{𝑚∣thereexistsaprojectiveresolution⋯→𝑃_𝑚+𝑗→⋯→𝑃_𝑚→𝑃_𝑚−1→⋯→𝑃₀→𝑀→0suchthat𝑃_𝑚+𝑖,𝑖=0,1,2,…arefinitelygenerated}. (2.1) If there is no such resolution, then define FPd(𝑀)=∞.

In particular, if FPd(𝑀)=0, then 𝑀 has an infinite finite presentation. In this case, we call 𝑀 a strongly presented module. Consequently, we may regard the presented dimension as a measure of how far away a module is from having infinite finite presentation.

Clearly, 𝑅 is right 𝑛-coherent if and only if FPd(𝑀)=0 for each (𝑛−1)-presented right 𝑅-module 𝑀, if and only if every (𝑛−1)-presented module has infinite finite presentation.

Proposition 2.2. Let 𝑀 be a right 𝑅-module, then FPd(𝑀)≤pd(𝑀)+1.

Proof. Directly by Definition 2.1.

We remark that FPd(𝑀) can be much smaller than pd(𝑀). Take 𝑅=ℤ₄. The ideal 2ℤ₄ has projective dimension ∞ while FPd(2ℤ₄)=0 for 𝑅 is Noetherian.

Proposition 2.3. No finitely generated right 𝑅-module has presented dimension 1.

Proof. Suppose that 𝑀 is a finitely generated right 𝑅-module with FPd(𝑀)=1. There is a projective resolution ⋯→𝑃₂→𝑃₁→𝑃−−→₀^𝑑0𝑀→0, (2.2) where 𝑃₁,𝑃₂,… are finitely generated; it follows that ker𝑑₀ is finitely presented and hence finitely generated. Note that 0→ker𝑑₀→𝑃₀→𝑀→0 is exact and 𝑀 is finitely generated, thus 𝑃₀ is finitely generated, so FPd(𝑀)=0, a contradiction.

It is known that every finitely presented flat right 𝑅-module is projective, that is, if f.p.dim(𝑀)=0, thenfd(𝑀)<f.p.dim(𝑀)+1⇒pd(𝑀)<f.p.dim(𝑀)+1. (2.3) For the presented dimensions of modules, we give a general result as follows.

Theorem 2.4. Assume that FPd(𝑀)=𝑚<∞ and 𝑡≥0 is an integer, then fd(M)<FPd(M)+𝑡iffpd(M)<FPd(M)+𝑡. (2.4)

Proof. When 𝑚=0, this is trivial. Now suppose that 0<𝑚<∞, then there is a projective resolution ⋯→𝑃−−−−→^𝑑_𝑚+𝑛^𝑚+𝑛⋯→𝑃−−→_𝑚^𝑑𝑚𝑃_𝑚−1→⋯→𝑃−−→₀^𝑑0𝑀→0, (2.5) where 𝑃_𝑚,…,𝑃_𝑚+𝑛,… are finitely generated.
We only need to prove the necessity. Let 𝐾_𝑖=ker𝑑_𝑖, 𝑖=0,1,…,𝑛+𝑚,…, and 𝐾₋₁=𝑀. For each integer 𝑡≥0, there is an exact sequence 0→𝐾_{𝑚+𝑡−2}→𝑃_{𝑚+𝑡−2}→⋯→𝑃_𝑚→⋯→𝑃₀→𝑀→0,0→𝐾_{𝑚+𝑡−1}→𝑃_{𝑚+𝑡−1}→𝐾_{𝑚+𝑡−2}→0. (2.6)(2.7) Since fd(𝑀)<𝑚+𝑡, we have that 𝐾_{𝑚+𝑡−2} is flat from (2.6). Note that 𝑃_𝑚+𝑡 is finitely generated and projective, thus 𝐾_{𝑚+𝑡−1}=Im𝑑_𝑚+𝑡 is finitely generated. From (2.7) and [7], it follows that 𝐾_{𝑚+𝑡−2} is projective. Thus pd(𝑀)≤𝑚+𝑡−1 from (2.6), so pd(𝑀)<𝑚+𝑡.

In particular, we have the following corollary.

Corollary 2.5. 𝑀 is a projective right 𝑅-module if and only if FPd(𝑀)≤1 and 𝑀 are flat.

Proof. (⇒). Immediately from Proposition 2.2.
(⇐). If FPd(𝑀)=0, then 𝑀 is finitely presented, Thus 𝑀 is projective.
If FPd(𝑀)=1, then fd(𝑀)<FPd(𝑀). From Theorem 2.4, we have pd(𝑀)<FPd(𝑀). Thus pd(𝑀)=0, so 𝑀 is projective.

We recall the mapping cone construction. Suppose that 𝐹∶𝐶^′→𝐶 is a morphism of complexes. Then 𝑀𝐶(𝐹) is a complex with 𝑀𝐶(𝐹)_𝑛=𝐶_𝑛⊕𝐶^′_𝑛−1, and the sequence of complexes 0→𝐶→𝑀𝐶(𝐹)→𝐶^′(−1)→0 is exact (see [8]).

Assume that(2.8)

is a commutative diagram in which the vertical maps are projective resolutions. If 𝑓 is a monomorphism, 𝑀𝐶(𝐹) is a projective resolution of coker𝑓. If 𝑓 is an epimorphism, then ⋯→𝑀𝐶(𝐹)_𝑛→𝑀𝐶(𝐹)_𝑛−1→⋯→𝑀𝐶(𝐹)₂→𝑍₁(𝑀𝐶(𝐹))→ker𝑓→0 (2.9) is a projective resolution of ker𝑓.

Assume that FPd(𝑀)=𝑚, there is an exact sequence ⋯→𝑃_𝑚+𝑛→⋯→𝑃_𝑚→𝑃_𝑚−1→⋯→𝑃₀→𝑀→0, (2.10) where 𝑃_𝑖(𝑖=0,…,𝑚+𝑛,…) are projective, and 𝑃_𝑚,…,𝑃_𝑚+𝑛,… are finitely generated; we call such an infinite exact sequence a representing sequence of 𝑀.

Theorem 2.6. Assume that 0→𝐴^′→𝐴→𝐴^′′→0 is an exact sequence of right 𝑅-modules, FPd(𝐴^′)=𝑑^′,FPd(𝐴)=𝑑,FPd(𝐴^′′)=𝑑^′′. If two of these are finite, then so is the third. Furthermore, 𝑑≤max{𝑑^′,𝑑^′′},𝑑^′′≤max{𝑑,𝑑^′+1},𝑑^′≤max{𝑑,𝑑^′′−1}. (2.11)

Proof. Suppose that 𝑑^′,𝑑^′′ are finite. Let 𝑃^′,𝑃^′′ represent sequences of 𝐴^′,𝐴^′′, respectively. There exists a projective resolution 𝑃 of 𝐴 such that 0→𝑃^′→𝑃→𝑃^′′→0 is an exact sequence of complexes. Thus 𝑃_𝑚 is finitely generated when 𝑚≥max{𝑑^′,𝑑^′′}. So FPd(𝐴)≤max{𝑑^′,𝑑^′′}.
Suppose that 𝑑^′,𝑑 are finite. Let 𝑃^′,𝑃 represent sequences of 𝐴^′,𝐴, respectively, and let 𝐹∶𝑃^′→𝑃 cover 𝑓∶𝐴^′→𝐴, thus 𝑀𝐶(𝐹) is a projective resolution of 𝐴^′′. By the definition of 𝑀𝐶(𝐹), we have that 𝑀𝐶(𝐹)_𝑚 is finitely generated for each 𝑚≥𝑑 and 𝑚≥𝑑^′+1. So FPd(𝐴^′′)≤max{𝑑,𝑑^′+1}.
Suppose that 𝑑,𝑑^′′ are finite. Let 𝑃,𝑃^′′ represent sequences of 𝐴,𝐴^′′, respectively, and let 𝐺∶𝑃→𝑃^′′ cover 𝑔∶𝐴→𝐴^′′. Then 𝑃^′ is a projective resolution of 𝐴^′, where 𝑃^′_𝑚=𝑀𝐶(𝐺)_𝑚+1(𝑚≥1), 𝑃^′₀=𝑍₁(𝑀𝐶(𝐺)). Thus 𝑀𝐶(𝐺)_𝑚 are finitely generated, whenever 𝑚≥𝑑^′′ and 𝑚≥𝑑+1. So 𝑃^′_𝑚 is finitely generated for 𝑚≥𝑑^′′−1 and 𝑚≥𝑑. Note that 𝑃^′₀ is finitely generated if 𝑑^′′≤1 and 𝑑=0 by the split exact sequence 0→𝑍₁(𝑀𝐶(𝐹))→𝑀𝐶(𝐹)₁→𝑀𝐶(𝐹)₀→0. (2.12) So FPd(𝐴^′)≤max{𝑑,𝑑^′′−1}.

Corollary 2.7. If FPd(𝐴₁),…,FPd(𝐴_𝑚) are finite, then FPd(𝐴₁⊕⋯⊕𝐴_𝑚)=max{FPd(𝐴_𝑖)∣𝑖=1,…,𝑚}. (2.13)

Proof. Clearly it suffices to consider the case 𝑚=2. Then there exist exact sequences 0→𝐴₁→𝐴₁⊕𝐴₂→𝐴₂→0,0→𝐴₂→𝐴₁⊕𝐴₂→𝐴₁→0. (2.14) By Theorem 2.6, we have FPd(𝐴₁⊕𝐴₂)≤max{FPd(𝐴₁),FPd(𝐴₂)},FPd(𝐴₁)≤max{FPd(𝐴₁⊕𝐴₂),FPd(𝐴₂)−1},FPd(𝐴₂)≤max{FPd(𝐴₁⊕𝐴₂),FPd(𝐴₁)−1}. (2.15) Suppose that FPd(𝐴₁⊕𝐴₂)<FPd(𝐴₁). Then FPd(𝐴₂)≤FPd(𝐴₁)−1, thus FPd(𝐴₁)≤max{FPd(𝐴₁⊕𝐴₂),FPd(𝐴₂)−1}≤max{FPd(𝐴₁⊕𝐴₂),FPd(𝐴₁)−2}=FPd(𝐴₁⊕𝐴₂), (2.16) which contradicts the hypothesis. So FPd(𝐴₁⊕𝐴₂)≥FPd(𝐴₁). Similarly, FPd(𝐴₁⊕𝐴₂)≥FPd(𝐴₂). Therefore FPd(𝐴₁⊕𝐴₂)=max{FPd(𝐴₁),FPd(𝐴₂)}.

3. Strongly Presented Modules

Theorem 3.1. FPd(𝑀)≤1 if and only if there are a projective module 𝑃₀, a free module 𝐹₀, and a strongly presented module 𝑀₀ such that 𝑀⊕𝑃₀=𝑀₀⊕𝐹₀.

Proof. (⇒). Suppose that FPd(𝑀)≤1. There is an exact sequence 0→𝐾→−^𝑖𝑃−→^𝜋𝑀→0, where 𝑃 is projective and 𝐾 is strongly presented. Choose a projective module 𝑃₀ such that 𝑃⊕𝑃₀ is free, and let 𝐹=𝑃⊕𝑃₀. Thus we have an exact sequence 0→𝐾→−^𝑖𝐹−−−−−−−→^{𝜎=𝜋⊕𝐼𝑃0}𝑀⊕𝑃₀→0. (3.1) Suppose that 𝐾 is generated by the set {𝑔₁,𝑔₂,…,𝑔_𝑛}. Choose a basis {𝑓₁,𝑓₂,…,𝑓_𝑗,…} of 𝐹 such that 𝑖(𝑔₁),𝑖(𝑔₂),…,𝑖(𝑔_𝑛) can be generated by 𝑓₁,𝑓₂,…,𝑓_𝑚. Let 𝐹₁ be generated by 𝑓₁,𝑓₂,…,𝑓_𝑚, and 𝐹₂ generated by 𝑓_𝑚+1,𝑓_𝑚+2,…,𝑓_𝑗,…. Then 𝐹=𝐹₁⊕𝐹₂, and 𝑖(𝐾)⊆𝐹₁. Let 𝑀₀=𝜎(𝐹₁), 𝐹₀=𝜎(𝐹₂). Then 𝑀₀ is strongly presented, 𝐹₀≅𝐹₂, and 𝑀⊕𝑃₀=𝑀₀⊕𝐹₀.
(⇐). Suppose that 𝑀⊕𝑃₀=𝑀₀⊕𝐹₀, where 𝑃₀ is a projective module, 𝐹₀ is a free module, and 𝑀₀ is a strongly presented module. There is a finitely generated free module 𝐹 such that the following sequence: 0→𝐾→−^𝑖𝐹₀⊕𝐹−→^𝑓𝐹₀⊕𝑀₀=𝑀⊕𝑃₀→0 (3.2) is exact and 𝐾 is strongly presented. Let 𝜋∶𝑀⊕𝑃₀→𝑃₀ be the canonical projection. Then we have an exact sequence 0→𝐾^′→𝐹₀⊕𝐹−−→^𝜋𝑓𝑃₀→0, where 𝐾^′=ker𝜋𝑓. It is clear that 𝑖(𝐾)⊆𝐾^′. Thus 0→𝐾→−^𝑖𝐾−−−→^{′𝑓|𝑘′}𝑀→0 is exact, hence ker𝑓|_𝐾′=ker𝑓∩𝐾^′=𝑖(𝐾)∩𝐾^′=𝑖(𝐾), (3.3) and 𝑓|_𝐾′ is epimorphic. Note that 𝐾 is strongly presented and 𝐾^′ is projective, thus FPd(𝑀)≤1.

Corollary 3.2. Assume that 𝑅 is a ring such that every projective module is free (e.g., 𝑅 is local). Then FPd(𝑀)≤1 if and only if there are a strongly presented module 𝑀₀ and a free module 𝐹 such that 𝑀=𝑀₀⊕𝐹.

Next, we aim to obtain a test for projectivity of modules with finite presented dimensions. In [1, Theorem 1.7], it was proved that pd(𝑀)≤𝑑 for every 𝑛-presented module 𝑀 if and only if Ext_𝑅^𝑑+1(𝑀,𝑁)=0 for every 𝑛-presented module 𝑀 and (𝑛−(𝑑+1))-presented module 𝑁. We generalize it as follows.

Proposition 3.3. Assume that 𝑀 is a strongly presented module and 𝑛≥0 is an integer. Then pd(𝑀)≤𝑛 if and only if Ext_𝑅^𝑛+1(𝑀,𝑁)=0 for every strongly presented module 𝑁.

Proof. The necessity is clear. Conversely, we proceed by induction on 𝑛. If 𝑛=0 and E𝑥𝑡¹_𝑅(𝑀,𝑁)=0 for every strongly presented module 𝑁, there is, an exact sequence 0→𝑀₁→𝐹→𝑀→0, (3.4) where 𝐹 is finitely generated and free and 𝑀₁ is strongly presented. Thus Ext¹_𝑅(𝑀,𝑀₁)=0, whence Hom(𝐹,𝑀₁)→Hom(𝑀₁,𝑀₁)→0; is exact, so 0→𝑀₁→𝐹→𝑀→0 is split, and 𝑀 is a direct summand of 𝐹, hence projective, that is, pd(𝑀)≤0.
Now suppose that 𝑛≥1 and E𝑥𝑡_𝑅^𝑛+1(𝑀,𝑁)=0 for every strongly presented module 𝑁. Since 0=Ext_𝑅^𝑛+1(𝑀,𝑁)=Ext^𝑛_𝑅(𝑀₁,𝑁), (3.5) and 𝑀₁ is strongly presented, by hypothesis pd(𝑀₁)≤𝑛−1, so pd(𝑀)≤𝑛.

Corollary 3.4. Assume that 𝑀 is strongly presented and 𝑛≥0 is an integer. If pd(𝑀)=𝑛, then Ext^𝑛_𝑅(𝑀,𝑅)≠0.

Proof. Since pd(𝑀)=𝑛, by Proposition 3.3 there is a strongly presented module 𝑁 such that Ext^𝑛_𝑅(𝑀,𝑁)≠0; thus there is an exact sequence 0→𝑁₁→𝑃→𝑁→0, where 𝑃 is finitely generated and projective, and 𝑁₁ is finitely generated. So we have an exact sequence Ext^𝑛_𝑅(𝑀,𝑃)→Ext^𝑛_𝑅(𝑀,𝑁)→0 for pd(𝑀)=𝑛.
Suppose that Ext^𝑛_𝑅(𝑀,𝑅)=0. Then Ext^𝑛_𝑅(𝑀,𝑃)=0 for each finitely generated projective module 𝑃, so Ext^𝑛_𝑅(𝑀,𝑁)=0, a contradiction. Therefore Ext^𝑛_𝑅(𝑀,𝑅)≠0.

Lemma 3.5. Assume that FPd(𝑀)≤1. Then pd(𝑀)≤𝑛 if and only if Ext_𝑅^𝑛+1(𝑀,𝑁)=0 for every strongly presented module 𝑁.

Proof. By Theorem 3.1, 𝑀⊕𝑃₀=𝑀₀⊕𝐹, where 𝑃₀ is projective, 𝐹 is free, and 𝑀₀ is strongly presented. Thus pd(𝑀)≤𝑛 if and only if Ext_𝑅^𝑛+1(𝑀,𝐵)=0 for every module 𝐵, if and only if Ext_𝑅^𝑛+1(𝑀₀,𝐵)=0 for every module 𝐵, if and only if pd(𝑀₀)≤𝑛, if and only if Ext_𝑅^𝑛+1(𝑀₀,𝑁)=0 for every strongly presented module 𝑁 by Proposition 3.3, if and only if Ext_𝑅^𝑛+1(𝑀,𝑁)=0 for every strongly presented module 𝑁.

Theorem 3.6. Assume that FPd(𝑀)<∞ and 𝑛 is an integer. Then pd(𝑀)≤𝑛 if and only if Ext_𝑅^𝑛+1(𝑀,𝑁)=0 for every strongly presented module 𝑁.

Proof. Suppose that FPd(𝑀)=𝑚. Then 𝑚≤pd(𝑀)+1 by Proposition 2.2, and there is a projective resolution of 𝑀⋯→𝑃−−−→^𝑑_𝑚+𝑙^𝑚+𝑙⋯→𝑃−−→_𝑚^𝑑𝑚𝑃_𝑚−1→⋯→𝑃−−→₀^𝑑0𝑀→0, (3.6) where 𝑃_𝑚,…,𝑃_𝑚+𝑙,… are finitely generated. Thus ker𝑑_𝑚−1 is strongly presented.
Suppose that 𝑛=𝑚−1. Then FPd(ker𝑑_𝑚−2)≤1, hence pd(ker𝑑_𝑚−2)≤𝑛−𝑚+1 if and only if Ext_𝑅^{𝑛−𝑚+2}(ker𝑑_𝑚−2,𝑁)=0 for every strongly presented module 𝑁 by Lemma 3.5. Suppose that 𝑛≥𝑚; by Proposition 3.3 pd(ker𝑑_𝑚−1)≤𝑛−𝑚 if and only if Ext_𝑅^{𝑛−𝑚+1}(ker𝑑_𝑚−1,𝑁)=0 for every strongly presented module 𝑁.
Therefore pd(𝑀)≤𝑛 if and only if E𝑥𝑡_𝑅^𝑛+1(𝑀,𝑁)=0 for every strongly presented module 𝑁.

Now we obtain a way to compute the right global dimension of a ring.

Corollary 3.7. Assume that rgD(R)<∞. Then rgD(𝑅)=sup{id(𝑁)∣𝑁isstronglypresented}. (3.7)

Proof. By Proposition 2.2.  FPd(𝑅/𝐼)<∞ for each right ideal 𝐼 of 𝑅, thus pd(𝑅/𝐼)≤𝑛 for each right ideal 𝐼 of 𝑅 if and only if Ext_𝑅^𝑛+1(𝑅/𝐼,𝑁)=0 for each strongly presented module 𝑁 and each right ideal 𝐼 of 𝑅 by Theorem 3.6, if and only if id(𝑁)≤𝑛 for each strongly presented module 𝑁 by the Baer Criterion for injectivity. Therefore the result holds.

4. Presented Dimensions of Rings

Definition 4.1. Define the presented dimension of 𝑅 as follows: FPD(𝑅)=sup{FPd(𝑀)∣𝑀isafinitelygeneratedright𝑅-module}. (4.1)

It is easy to see that FPD(𝑅)=0 if and only if every finitely generated module has an infinite finite presentation, if and only if every finitely generated module is finitely presented, if and only if 𝑅 is right Noetherian. Thus we may regard the presented dimension of a ring as a measure of how far it is from being right Noetherian.

Proposition 4.2. FPD(𝑅)≤rgD(𝑅)+1.

Proof. By Proposition 2.2, FPD(𝑀)≤pd(𝑀)+1, thus the result follows immediately.

Note that FPD(𝑅) can be much smaller than rgD(𝑅). Take 𝑅=ℤ₄. Then rgD(𝑅)=∞ while FPD(𝑅)=0 for 𝑅 is Noetherian.

Following Proposition 2.3, we have the following corollary.

Corollary 4.3. No ring can have presented dimension 1.

In the following, we investigate the relations of the right global, weak global, and presented dimensions of rings.

Theorem 4.4. Let 𝑅 be a ring. (1)If  FPD(𝑅)≤wD(𝑅), then rgD(𝑅)=wD(𝑅).(2)If  FPD(𝑅)>wD(𝑅), then rgD(𝑅)=FPD(𝑅) or FPD(𝑅)−1.(3)If  rgD(𝑅)>wD(𝑅), then FPD(𝑅)=rgD(𝑅)+1.

Proof. (1) It suffices to prove that wD(𝑅)≥rgD(𝑅) and suppose that wD(𝑅)=𝑠<∞. Let 𝑀 be finitely generated. Since FPD(𝑅)≤wD(𝑅)=𝑠, we have FPd(𝑀)=𝑚≤𝑠, thus there is a projective resolution ⋯→𝑃−−−−→^𝑑_𝑚+𝑛^𝑚+𝑛⋯→𝑃−−→_𝑚^𝑑𝑚⋯→𝑃₀→𝑀→0, (4.2) where 𝑃_𝑚,…,𝑃_𝑚+𝑛,… are finitely generated. Since wD(𝑅)=𝑠, it follows that ker𝑑_𝑠−1 is flat. Note that 𝑠≥𝑚, hence ker𝑑_𝑠−1 is finitely presented, whence projective, that is, pd(𝑀)≤𝑠. So wD(𝑅)≥rgD(𝑅).
(2) If FPD(𝑅)=∞, the result follows immediately from Proposition 4.2. Now suppose that FPD(𝑅)=𝑚<∞. Since FPD(𝑅)>wD(𝑅)≥0, by Corollary 4.3 we have 𝑚≥2. Let 𝑀 be finitely generated and fd(𝑀)=𝑘, thus ker𝑑_𝑡 is finitely presented for each 𝑡≥𝑚−1.
If fd(𝑀)≤FPd(𝑀), then ker𝑑_𝑚−1 is flat, hence projective, so pd(𝑀)≤FPd(𝑀).
If fd(𝑀)>FPd(𝑀), then ker𝑑_𝑘−1 is flat, hence projective, so pd(𝑀)≤fd(𝑀)≤wD(𝑅). (4.3)
Therefore rgD(𝑅)≤FPD(𝑅).
On the other hand, by Proposition 4.2, FPD(𝑅)≤rgD(𝑅)+1. So rgD(𝑅)=FPD(𝑅) or FPD(𝑅)−1.
(3) From (1) and (2), we have FPD(R)=rgD(𝑅)+1 or FPD(𝑅)=rgD(𝑅). Thus we need only to consider rgD(𝑅)=𝑚<∞ and prove FPD(𝑅)≠rgD(𝑅). Suppose that FPD(𝑅)=rgD(𝑅). Let 𝑀 be a finitely generated right 𝑅-module with FPd(𝑀)=𝑚, then there is an exact sequence 0→𝑃_𝑚→𝑃−−−−→^𝑑_𝑚−1^𝑚−1𝑃−−−−→^𝑑_𝑚−2^𝑚−2⋯→𝑃₀→𝑀→0, (4.4) where 𝑃_𝑖 is projective and 𝑃_𝑚 is strongly presented. Let 𝐾_𝑚−2=ker𝑑_𝑚−2. Note that 𝑚≠1 and 𝑚>wD(𝑅)≥0. We consider the exact sequence 0→𝑃_𝑚→𝑃_𝑚−1→𝐾_𝑚−2→0. (4.5) Since wD(𝑅)<rgD(𝑅)=𝑚, 𝐾_𝑚−2 is flat. Suppose that 𝑄 such that 𝑃_𝑚−1⊕𝑄=𝐹 is free. Then 0→𝑃_𝑚→𝐹→𝐾_𝑚−2⊕𝑄→0 (4.6) is exact, and 𝐾_𝑚−2⊕𝑄 is flat. Let 𝑝₁,…,𝑝_𝑚 generate 𝑃_𝑚. Using the flatness of 𝐾_𝑚−2⊕𝑄, there exists a homomorphism 𝐹→𝑃_𝑚 such that 𝑝_𝑖↦𝑝_𝑖(𝑖=1,2,…,𝑡). Thus the above short sequence splits, and so 𝐹≅𝑃_𝑚⊕𝐾_𝑚−2⊕𝑄. Thus 𝐾_𝑚−2 is projective, therefore pd(𝑀)≤𝑚−1, and so rgD(𝑅)≤𝑚−1, a contradiction. Hence FPD(𝑅)≠rgD(𝑅), so FPD(𝑅)=rgD(𝑅)+1.

Corollary 4.5. rgD(𝑅)=max{wD(𝑅),FPD(𝑅)−1}.

From the foregoing discussion, we can classify rings by the right global dimensions, weak global dimensions, and presented dimensions of rings.

In the diagrams, represents two consecutive numbers while means that the numbers may not be consecutive.

5. On Ring Extensions

In this section, assume that 𝑆≥𝑅 is a unitary ring extension. We aim to investigate properties of the presented dimensions of modules and rings. We first recall some concepts.

(1) The ring 𝑆 is called right 𝑅-projective [9] in case, for any right 𝑆-module 𝑀_𝑆 with an 𝑆-module 𝑁_𝑆, 𝑁_𝑅∣𝑀_𝑅 implies 𝑁_𝑆∣𝑀_𝑆, where 𝑁∣𝑀 means that 𝑁 is a direct summand of 𝑀. For example, every 𝑛×𝑛 matrix ring 𝑅_𝑛 is right 𝑅-projective [9].

(2) The ring extension 𝑆≥𝑅 is called a finite normalizing extension [10] in case there is a finite subset {𝑠₁,…,𝑠_𝑛}⊆𝑆 such that 𝑆=∑^𝑛_𝑖=1𝑠_𝑖𝑅 and 𝑠_𝑖𝑅=𝑅𝑠_𝑖 for 𝑖=1,…,𝑛.

(3) A finite normalizing extension 𝑆≥𝑅 is called an almost excellent extension [11] in case _𝑅𝑆 is flat, 𝑆_𝑅 is projective, and the ring 𝑆 is right 𝑅-projective.

(4) An almost excellent extension 𝑆≥𝑅 is an excellent extension [9] in case both _𝑅𝑆 and 𝑆_𝑅 are free modules with a common basis {𝑠₁,…,𝑠_𝑛}.

Excellent extensions were introduced by Passman [9] and named by Bonami [12]. Examples include the 𝑛×𝑛 matrix rings and the crossed products 𝑅∗𝐺 where 𝐺 is a finite group with |𝐺|⁻¹∈𝑅. Almost excellent extensions were introduced and studied by Xue [11] as a nontrivial generalization of excellent extensions and recently studied in [2, 13–15].

Proposition 5.1. Assume that 𝑆≥𝑅 is a finite normalizing extension and _𝑅𝑆 is flat. Then for each right 𝑅-module 𝑀_𝑅, we have FPd(𝑀⊗_𝑅𝑆)_𝑆≤FPd(𝑀_𝑅). (5.1)

Proof. If FPd(𝑀_𝑅)=∞, it is clear. Suppose FPd(𝑀_𝑅)=𝑚<∞. There is a projective resolution of 𝑀⋯→𝑃_𝑚+𝑛→⋯→𝑃_𝑚→𝑃_𝑚−1→⋯→𝑃₀→𝑀→0, (5.2) where 𝑃_𝑚,…,𝑃_𝑚+𝑛 are finitely generated. Since _𝑅𝑆 is flat, there is an exact sequence of right 𝑆-modules ⋯→𝑃_𝑚+𝑛⊗_𝑅𝑆→⋯→𝑃_𝑚⊗_𝑅𝑆→⋯→𝑃₀⊗_𝑅𝑆→𝑀⊗_𝑅𝑆→0, (5.3) where 𝑃_𝑖⊗_𝑅𝑆 is a projective right 𝑆-module, and 𝑃_𝑚⊗_𝑅𝑆,…,𝑃_𝑚+𝑛⊗_𝑅𝑆,… are finitely generated. So FPd(𝑀⊗_𝑅𝑆)≤𝑚, therefore FPd(𝑀⊗_𝑅𝑆)_𝑆≤FPd(𝑀_𝑅).

Proposition 5.2. Assume that 𝑆≥𝑅 is a finite normalizing extension, _𝑅𝑆 is flat, and 𝑆 is right 𝑅-projective. Then for each right 𝑆-module 𝑀_𝑆, one has FPd(𝑀_𝑆)≤FPd(𝑀⊗_𝑅𝑆). (5.4)

Proof. By [11, Lemma 1.1], 𝑀_𝑆 is isomorphic to a direct summand of 𝑀⊗_𝑅𝑆. By Corollary 2.7, FPd(𝑀⊗_𝑅𝑆)≥FPd(𝑀_𝑆).

Proposition 5.3. Assume that 𝑆≥𝑅 is an almost excellent extension. Then for each right 𝑆-module 𝑀_𝑆, one has FPd(𝑀_𝑅)≤FPd(𝑀_𝑆).

Proof. If FPd(𝑀_𝑆)=∞, then it clear. Suppose that FPd(𝑀_𝑆)=𝑚<∞. Then there is a projective resolution ⋯→𝑃_𝑚+𝑛→⋯→𝑃_𝑚⋯→𝑃₀→𝑀_𝑆→0, (5.5) where 𝑃_𝑖(𝑖=0,1,…) are right 𝑆-modules and 𝑃_𝑚,…,𝑃_𝑚+1,… are finitely generated. Since 𝑆≥𝑅 is an almost excellent extension, it follows that 𝑃_𝑖(𝑖=0,1,…) are projective right 𝑅-modules, and 𝑃_𝑚,𝑃_𝑚+1,… are finitely generated right 𝑅-modules. Thus ⋯→𝑃_𝑚+𝑛→⋯→𝑃_𝑚→𝑃_𝑚−1→⋯→𝑃₀→𝑀→0, (5.6) is a projective resolution of 𝑀_𝑅. So FPd(𝑀_𝑆)≥FPd(𝑀_𝑅).

Corollary 5.4. Assume that 𝑆≥𝑅 is an almost excellent extension. Then for each right 𝑆-module 𝑀_𝑆, one has FPd(𝑀_𝑅)=FPd(𝑀_𝑆)=FPd(𝑀⊗_𝑅𝑆).

Theorem 5.5. Assume that 𝑆≥𝑅 is a finite normalizing extension and _𝑅𝑆 is flat. (1)If 𝑆 is right 𝑅-projective and FPD(𝑆)<∞, then FPD(𝑆)≤FPD(𝑅);(2)If FPD(𝑅)<∞, then FPD(𝑅)≤FPD(𝑆)+max{𝑙,𝑠}, (5.7) where 𝑙=pd(𝑆_𝑅) and 𝑠=sup{FPd(𝐴_𝑅)∣𝐴∈Mod-𝑆andFPd(𝐴_𝑆)=0}.

Proof. (1) Suppose that FPD(𝑆)=𝑚, 𝑀 is a finitely generated right 𝑆-module, and FPd(𝑀_𝑆)=𝑚. Since 𝑆 is right 𝑅-projective, there is an exact sequence of 𝑆-modules 0→𝑀_𝑆→𝑀⊗_𝑅𝑆→𝐶→0, (5.8) where FPd(𝐶)≤𝑚 for FPD(𝑆)=𝑚. By Theorem 2.6 we have FPd(𝑀⊗_𝑅𝑆)_𝑆≤max{FPd(𝑀_𝑆),FPd(𝐶_𝑆)}=𝑚,𝑚=FPd(𝑀_𝑆)≤max{FPd(𝑀⊗_𝑅𝑆)_𝑆,FPd(𝐶_𝑆)−1}≤FPd(𝑆)=𝑚, (5.9) thus FPd(𝑀⊗_𝑅𝑆_𝑆)=𝑚. It follows from Proposition 5.1 that FPd(𝑀⊗_𝑅𝑆_𝑆)≤FPd(𝑀_𝑅). So FPD(𝑆)≤FPD(𝑅).
(2) Suppose that FPD(𝑅)=𝑚, 𝑀 is a finitely generated right 𝑅-module, and FPd(𝑀_𝑅)=𝑚<∞. Since _𝑅𝑆 is flat, by [16, Lemma 2.3], there is an exact sequence of 𝑅-modules0→𝑀→𝑀⊗_𝑅𝑆→𝐶→0. (5.10) Note that _𝑅𝑆 is finitely generated, which implies thats 𝑀⊗_𝑅𝑆 and 𝐶 are finitely generated, thus FPd(𝐶)≤𝑚 for FPD(𝑅)=𝑚. By Theorem 2.6, we have 𝑚=FPd(𝑀_𝑅)≤max{FPd(𝑀⊗_𝑅𝑆_𝑅),FPd(𝐶_𝑅)−1}≤FPD(𝑅)=𝑚, (5.11) hence FPd(𝑀⊗_𝑅𝑆_𝑅)=𝑚. Let FPd(𝑀⊗_𝑅𝑆)_𝑆=𝑡≤FPD(𝑆). Then there is a projective resolution of the right 𝑆-module 𝑀⊗_𝑅𝑆⋯→𝑄_𝑡+1→𝑄_𝑡→𝑄_𝑡−1→⋯→𝑄₀→𝑀⊗_𝑅𝑆→0, (5.12) where 𝑄_𝑡,𝑄_𝑡+1,… are finitely generated. Thus we have the following exact sequences: 0→𝐾_𝑡−1→𝑄_𝑡−1→𝐾_𝑡−2→0,0→𝐾_𝑡−2→𝑄_𝑡−2→𝐾_𝑡−3⋯→0,0→𝐾₀→𝑄₀→𝑀⊗_𝑅𝑆→0, (5.13) where 𝐾_𝑖=Im(𝑄_𝑖+1→𝑄_𝑖) and FPd(𝐾_𝑡−1)=0. By Proposition 2.2, FPd(𝑄_𝑖)_𝑅≤pd(𝑄_𝑖)_𝑅+1≤pd(𝑆_𝑅)+1=𝑙+1, (5.14) and FPd(𝐾_𝑡−1)_𝑅≤𝑠. Following Theorem 2.6 and Proposition 2.2, we have FPd_𝑅(𝐾_𝑡−2)≤max{FPd_𝑅(𝑄_𝑡−1),FPd_𝑅(𝐾_𝑡−1)+1}≤max{𝑙+1,𝑠+1}=1+max{𝑙,𝑠}. (5.15) Again by Theorem 2.6, we have FPd_𝑅(𝐾_𝑡−3⋯)≤2+max{𝑙,𝑠},FPd_𝑅(𝐾₀)≤𝑡−1+max{𝑙,𝑠},𝑚=FPd(𝑆⊗_𝑅𝑀)_𝑅≤𝑡+max{𝑙,𝑠}≤FPD(𝑆)+max{𝑙,𝑠}. (5.16) Therefore FPD(𝑅)≤FPD(𝑆)+max{𝑙,𝑠}.

Note that if 𝑆≥𝑅 is an almost excellent extension, then pd(𝑆_𝑅)=0, and 𝑠=sup{FPd(𝐴_𝑅)∣𝐴∈Mod-𝑆andFPd(𝐴_𝑆)=0}=0. (5.17)

thus

Corollary 5.6. Assume that 𝑆≥𝑅 is an almost excellent extension. Then FPD(𝑅)=FPD(𝑆).

Proof. Suppose that 𝑆≥𝑅 is an almost excellent extension. Then 𝑆_𝑅 is a finitely generated projective 𝑅-module and 𝑆 is right 𝑅-projective. By Theorem 5.5 and Proposition 5.2, we have FPD(𝑅)=FPD(𝑆).

To close this section, we give an example of an excellent extension 𝑆≥𝑅, which is provided by Xue in [16]. Let 𝑅 be a ring graded by a finite group 𝐺. The smash product 𝑅#𝐺 is a free right and left 𝑅-module with a basis {𝑝_𝑔∣𝑔∈𝐺} and the multiplication determined by (𝑟𝑝_𝑔)(𝑟^′𝑝_ℎ)=𝑟𝑟^′_𝑔ℎ−1𝑝ℎ, where 𝑔,ℎ∈𝐺,𝑟,𝑟^′∈𝑅, and 𝑟^′_𝑔ℎ−1 is the 𝑔ℎ⁻¹-component of 𝑟^′.

Example 5.7. Let 𝑅 be a ring graded by a finite group 𝐺 with |𝐺|⁻¹∈𝑅. Then FPD(𝑅)=FPD(𝑅#𝐺). (5.18)

Proof. By [17, Theorem 1.3], we know that 𝐺 acts as automorphisms on 𝑅#𝐺 and the skew group ring (𝑅#𝐺)∗𝐺≅𝑅_𝑛 where 𝑛=|𝐺|. Since skew group rings and finite matrix rings are excellent extensions, the result follows.

Acknowledgments

The authors would like to thank the referees for careful reading and correcting. This research is supported by the Natural Science Foundation of Fujian Province (2009J01003) and the Foundation of Fujian Normal University (2008100209, 09A004).