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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 256267, 13 pages
http://dx.doi.org/10.1155/2010/256267
Research Article

On Presented Dimensions of Modules and Rings

1Department of Mathematics, Fujian Normal University, Fuzhou 350007, China
2College of Computer and Information, Fujian Agriculture and Forestry University, Fuzhou 350002, China

Received 21 August 2009; Accepted 12 March 2010

Academic Editor: Francois Goichot

Copyright © 2010 Dexu Zhou and Zhiwei Gong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 be a ring and a nonnegative integer. Following [1, 2], a right -module is called -presented in case it has a finite -presentation, that is, there is an exact sequence of right -modules where each is a finitely generated free, equivalently projective, right -module. A module is 0-presented (resp., 1-presented) if and only if it is finitely generated (resp., finitely presented), and each -presented module is -presented for . A ring is called right -coherent in case every -presented right -module is -presented. It is easy to see that is right 0-coherent (resp., 1-coherent) if and only if it is right Noetherian (resp., coherent), and every -coherent ring is -coherent for .

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

Ng [4] defined the finitely presented dimension of a module as = inf there exists an exact sequence of -modules, where each is projective, and 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 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 -modules, and homomorphisms are module homomorphisms. The notations , denote the projective, injective, flat dimension of , and , 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 be a right -module, define the presented dimension of as follows: If there is no such resolution, then define .

In particular, if , then has an infinite finite presentation. In this case, we call 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, is right -coherent if and only if for each -presented right -module , if and only if every -presented module has infinite finite presentation.

Proposition 2.2. Let be a right -module, then .

Proof. Directly by Definition 2.1.

We remark that can be much smaller than . Take . The ideal has projective dimension while for is Noetherian.

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

Proof. Suppose that is a finitely generated right -module with . There is a projective resolution where are finitely generated; it follows that is finitely presented and hence finitely generated. Note that is exact and is finitely generated, thus is finitely generated, so , a contradiction.

It is known that every finitely presented flat right -module is projective, that is, if then For the presented dimensions of modules, we give a general result as follows.

Theorem 2.4. Assume that and is an integer, then

Proof. When , this is trivial. Now suppose that , then there is a projective resolution where are finitely generated.
We only need to prove the necessity. Let , , and . For each integer , there is an exact sequence Since , we have that is flat from (2.6). Note that is finitely generated and projective, thus is finitely generated. From (2.7) and [7], it follows that is projective. Thus from (2.6), so .

In particular, we have the following corollary.

Corollary 2.5. is a projective right -module if and only if and are flat.

Proof. Immediately from Proposition 2.2.
If , then is finitely presented, Thus is projective.
If , then . From Theorem 2.4, we have . Thus , so is projective.

We recall the mapping cone construction. Suppose that is a morphism of complexes. Then is a complex with , and the sequence of complexes is exact (see [8]).

Assume that256267.xy.001(2.8)

is a commutative diagram in which the vertical maps are projective resolutions. If is a monomorphism, is a projective resolution of . If is an epimorphism, then is a projective resolution of .

Assume that , there is an exact sequence where are projective, and are finitely generated; we call such an infinite exact sequence a representing sequence of .

Theorem 2.6. Assume that is an exact sequence of right -modules, . If two of these are finite, then so is the third. Furthermore,

Proof. Suppose that are finite. Let represent sequences of , respectively. There exists a projective resolution of such that is an exact sequence of complexes. Thus is finitely generated when . So .
Suppose that are finite. Let represent sequences of , respectively, and let cover , thus is a projective resolution of . By the definition of , we have that is finitely generated for each and . So .
Suppose that are finite. Let represent sequences of , respectively, and let cover . Then is a projective resolution of , where , . Thus are finitely generated, whenever and . So is finitely generated for and . Note that is finitely generated if and by the split exact sequence So .

Corollary 2.7. If are finite, then

Proof. Clearly it suffices to consider the case . Then there exist exact sequences By Theorem 2.6, we have Suppose that . Then , thus which contradicts the hypothesis. So . Similarly, . Therefore .

3. Strongly Presented Modules

Theorem 3.1. if and only if there are a projective module , a free module , and a strongly presented module such that .

Proof. Suppose that . There is an exact sequence , where is projective and is strongly presented. Choose a projective module such that is free, and let . Thus we have an exact sequence Suppose that is generated by the set . Choose a basis of such that can be generated by . Let be generated by , and generated by . Then , and . Let , . Then is strongly presented, , and .
Suppose that , where is a projective module, is a free module, and is a strongly presented module. There is a finitely generated free module such that the following sequence: is exact and is strongly presented. Let be the canonical projection. Then we have an exact sequence where . It is clear that . Thus is exact, hence and is epimorphic. Note that is strongly presented and is projective, thus .

Corollary 3.2. Assume that is a ring such that every projective module is free e.g., is local. Then if and only if there are a strongly presented module and a free module such that .

Next, we aim to obtain a test for projectivity of modules with finite presented dimensions. In [1, Theorem ], it was proved that pd for every -presented module if and only if for every -presented module and -presented module . We generalize it as follows.

Proposition 3.3. Assume that is a strongly presented module and is an integer. Then if and only if for every strongly presented module .

Proof. The necessity is clear. Conversely, we proceed by induction on . If and for every strongly presented module , there is, an exact sequence where is finitely generated and free and is strongly presented. Thus , whence is exact, so is split, and is a direct summand of , hence projective, that is, pd.
Now suppose that and for every strongly presented module . Since and is strongly presented, by hypothesis pd, so pd.

Corollary 3.4. Assume that is strongly presented and is an integer. If pd, then .

Proof. Since pd, by Proposition 3.3 there is a strongly presented module such that ; thus there is an exact sequence where is finitely generated and projective, and is finitely generated. So we have an exact sequence for pd.
Suppose that . Then for each finitely generated projective module , so , a contradiction. Therefore .

Lemma 3.5. Assume that . Then pd if and only if for every strongly presented module .

Proof. By Theorem 3.1, , where is projective, is free, and is strongly presented. Thus pd if and only if for every module , if and only if for every module , if and only if pd, if and only if for every strongly presented module by Proposition 3.3, if and only if for every strongly presented module .

Theorem 3.6. Assume that and is an integer. Then pd if and only if for every strongly presented module .

Proof. Suppose that . Then by Proposition 2.2, and there is a projective resolution of where are finitely generated. Thus is strongly presented.
Suppose that . Then , hence pd if and only if for every strongly presented module by Lemma 3.5. Suppose that ; by Proposition 3.3 pd if and only if for every strongly presented module .
Therefore pd if and only if for every strongly presented module .

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

Corollary 3.7. Assume that . Then

Proof. By Proposition 2.2.   for each right ideal of , thus pd for each right ideal of if and only if for each strongly presented module and each right ideal of by Theorem 3.6, if and only if for each strongly presented module by the Baer Criterion for injectivity. Therefore the result holds.

4. Presented Dimensions of Rings

Definition 4.1. Define the presented dimension of as follows:

It is easy to see that 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 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. .

Proof. By Proposition 2.2, , thus the result follows immediately.

Note that can be much smaller than . Take . Then while for 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 be a ring. (1)If  , then .(2)If  , then or .(3)If  , then .

Proof. () It suffices to prove that and suppose that . Let be finitely generated. Since , we have , thus there is a projective resolution where are finitely generated. Since , it follows that is flat. Note that , hence is finitely presented, whence projective, that is, . So .
() If , the result follows immediately from Proposition 4.2. Now suppose that . Since , by Corollary 4.3 we have . Let be finitely generated and , thus is finitely presented for each .
If , then is flat, hence projective, so .
If , then is flat, hence projective, so
Therefore .
On the other hand, by Proposition 4.2, . So or .
(3) From () and (), we have or . Thus we need only to consider and prove . Suppose that . Let be a finitely generated right -module with , then there is an exact sequence where is projective and is strongly presented. Let . Note that and We consider the exact sequence Since , is flat. Suppose that such that is free. Then is exact, and is flat. Let generate . Using the flatness of , there exists a homomorphism such that . Thus the above short sequence splits, and so . Thus is projective, therefore , and so , a contradiction. Hence , so .

Corollary 4.5. .

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

256267.fig.001

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 is a unitary ring extension. We aim to investigate properties of the presented dimensions of modules and rings. We first recall some concepts.

() The ring is called right -projective [9] in case, for any right -module with an -module , implies , where means that is a direct summand of . For example, every matrix ring is right -projective [9].

() The ring extension is called a finite normalizing extension [10] in case there is a finite subset such that and for .

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

() An almost excellent extension is an excellent extension [9] in case both and are free modules with a common basis .

Excellent extensions were introduced by Passman [9] and named by Bonami [12]. Examples include the matrix rings and the crossed products where is a finite group with . Almost excellent extensions were introduced and studied by Xue [11] as a nontrivial generalization of excellent extensions and recently studied in [2, 1315].

Proposition 5.1. Assume that is a finite normalizing extension and is flat. Then for each right -module , we have

Proof. If , it is clear. Suppose . There is a projective resolution of where are finitely generated. Since is flat, there is an exact sequence of right -modules where is a projective right -module, and are finitely generated. So therefore

Proposition 5.2. Assume that is a finite normalizing extension, is flat, and is right -projective. Then for each right -module , one has

Proof. By [11, Lemma ], is isomorphic to a direct summand of . By Corollary 2.7,

Proposition 5.3. Assume that is an almost excellent extension. Then for each right -module , one has

Proof. If , then it clear. Suppose that . Then there is a projective resolution where are right -modules and are finitely generated. Since is an almost excellent extension, it follows that are projective right -modules, and are finitely generated right -modules. Thus is a projective resolution of . So .

Corollary 5.4. Assume that is an almost excellent extension. Then for each right -module , one has

Theorem 5.5. Assume that is a finite normalizing extension and is flat. (1)If is right -projective and , then (2)If , then where and .

Proof. () Suppose that , is a finitely generated right -module, and . Since is right -projective, there is an exact sequence of -modules where for . By Theorem 2.6 we have thus . It follows from Proposition 5.1 that . So .
() Suppose that , is a finitely generated right -module, and . Since is flat, by [16, Lemma ], there is an exact sequence of -modules Note that is finitely generated, which implies thats and are finitely generated, thus for . By Theorem 2.6, we have hence . Let . Then there is a projective resolution of the right -module where are finitely generated. Thus we have the following exact sequences: where and . By Proposition 2.2, and . Following Theorem 2.6 and Proposition 2.2, we have Again by Theorem 2.6, we have Therefore .

Note that if is an almost excellent extension, then , and

thus

Corollary 5.6. Assume that is an almost excellent extension. Then

Proof. Suppose that is an almost excellent extension. Then is a finitely generated projective -module and is right -projective. By Theorem 5.5 and Proposition 5.2, we have

To close this section, we give an example of an excellent extension , which is provided by Xue in [16]. Let be a ring graded by a finite group . The smash product is a free right and left -module with a basis and the multiplication determined by , where , and is the -component of .

Example 5.7. Let be a ring graded by a finite group with . Then

Proof. By [17, Theorem ], we know that acts as automorphisms on and the skew group ring where . 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).

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