Abstract

Matrix factorization has been widely investigated in the past years due to its fundamental importance in several areas of engineering. This paper investigates completion and zero prime factorization of matrices over elementary divisor rings (EDR). The Serre problem and Lin-Bose problems are generalized to EDR and are completely solved.

1. Introduction

In engineering and communication sciences, polynomial matrices are used in several different areas including circuits, multidimensional systems, controls, signal processing, and other areas. The Serre problem (or Serre Theorem) stands for a fundamental breakthrough in the understanding of polynomial matrices, and it is a powerful mathematical tool for engineers in practical designs. Following the work of Youla and Gnavi [1] on the basic structure of -D system theory, many papers have been published in studying various prime factorization of multivariate polynomial matrices.

Lin and Bose in 2001 [2] formulated a generalized Serre conjecture for the polynomial ring over a field . They found out that zero prime matrix completion and matrix primitive factorization were all related to the generalized Serre conjecture. So they proposed the existence problem of zero prime factorization for -D polynomial matrices, which is now called Lin-Bose problem and it has been solved in [3–5].

We are interested in generalizing Serre conjecture and Lin-Bose problem to elementary divisor rings, which is defined in the next section. For example, let be any row vector with entries in , and let be any maximal common divisor of . We want to know if the row can be completed to a square matrix whose determinant is . More generally, we will solve both Serre problem and Lin-Bose problem for an arbitrary matrix (not just a row) over elementary divisor rings.

The organization of the paper is as follows. In Section 2, we first give some basic notions and describe Serre Problem and Lin-Bose problem precisely. In Section 3, we give proofs for the problems proposed in Section 2. Finally, a brief conclusion is given in Section 4.

2. Basic Notions and Main Problems

Let be a commutative ring with a unity element and the free module of matrices with entries in . For any , denotes the ideal of generated by all minors of , where . Set . The rank of is defined to be where When , we say is of full row rank.

Definition 1. A commutative ring is called an elementary divisor ring (EDR) if, for every and every matrix , there exist and such that with is diagonal and every divides .

When , is row vector ; the requirement implies that the ideal is generated by one element. So, in an elementary divisor ring, every finitely generated ideal is generated by one element. Note that an elementary divisor ring may not be a principle ideal domain, nor a unique factorization domain.

Definition 2. Let (or ), where , be of full row rank. Then is said to be (i)ZLP (or ZRP) if all minors of generate the unit ideal ;(ii)MLP if all minors of are relatively prime; that is, is a unit in , where refer to the maximal common divisor of all minors of .

Definition 3. Let . is said to be a common divisor of if , . When is divisible by every common divisor of , one says that is a maximal common divisor of .

Note that, for any two maximal common divisors of , since they divide each other, they are always associates of each other; that is, they are different only by a factor that is invertible in . Let denote the set of all maximal common divisors of .

Lemma 4. Let be an elementary divisor ring and . For any , one has if and only if .

Proof. First suppose . Since has a unity element , we have , so for . Also, for some ; hence, for any such that for , we must have . Thus .
Next suppose . Then for some , , so . Since is an EDR, there exists such that . This implies that , , so ; thus and . Therefore, .

A direct consequence of the above lemma is that, in an elementary divisor ring, any collection of elements has at least one maximal common divisor, since the ideal is generated by one element. This means that, for a unimodular row , the maximal common divisors of must be units.

Definition 5. Let with , and let denote all minors of , where . Assume that there exists a maximal common divisor of . Let be such that , . Then are called the “reduced minors” of with respect to .

The original Serre problem and Lin-Bose problems are about the ring , a polynomial ring in the variable over a field . More precisely, for any () of full row rank, let be the greatest common divisor of all minors of . Suppose all reduced minors of generate . Then Serre’s problem says that there exists a matrix such that . Lin-Bose problem says that we can decompose as , where , , , and is ZLP.

In this paper, we extend the above two problems over to elementary divisor rings. Precisely, we completely solve the following problems.

Problem 6. Let be an elementary divisor ring and , where , is of full row rank. Let be a maximal common divisors of all minors of . (a)(Serre) Is there a matrix such that ?(b)(Lin-Bose) Is it possible to write as , where with and is ZLP?

3. Main Results

In this section, we give our main results. First, let us give some basic facts. For more details, we refer to [6]. Let be a commutative ring. Then any finite number of elements in have a maximal common divisor. Let be of full row rank. Let ’s be all of its minors and the maximal common divisor of ’s. Then there exists a matrix such that , where and is the identity matrix. Furthermore, if all the reduced minors of generate the unit ideal , then there exists a matrix such that . When , we have .

Lemma 7. Let be an EDR. Let with with being a maximal common divisor of all minors of . Then, for every , the matrix also has as a maximal common divisor of its minors.

Proof. Let be any maximal divisor of all minors of with . Let be any submatrix of . Then , where is a submatrix of . Then, by Cauchy-Binet formula, we can get that , where and are minors of and , respectively. Since, for every , we have and , by the arbitrariness of , we get . Since , by the same reason, we have , so . Therefore, is also a maximal divisor of minors of .

Theorem 8. Let be an EDR and . Let be an arbitrary row of and any maximal common divisor of . Then can be completed to a square matrix whose determinant is . Furthermore, the matrix may be chosen to be itself completed to a matrix in .

Proof. Assume without loss of generality that is the first row of ; then according to the definition of an elementary divisor ring, there exist and such that By Lemma 4   is a maximal common divisor of . Assume that and are units in . LetThen, , , , and where is the submatrix of formed by the remaining rows after removing from .
Set . Then Note that . Thus So is ZLP and can be completed to matrix in .
By Lemma 4, ; we have that and , where . Then This proves the theorem.

In the above theorem, is an arbitrary maximal common divisor, but is not UFD so the maximal common divisors are not unique. If is a beforehand given maximal common divisor, is the above theorem also correct? The following theorem gives a positive answer.

Theorem 9. Let be an EDR and with . Then there exist and such that , where is a maximal common divisor of all minors of and is ZLP.

Proof. Since is an elementary divisor ring, there exist and such that and every is a divisor of , and . By Lemma 7, is a maximal common divisor of all minors of . Partition as Let . Since , by Laplace expansion, is ZLP. Since , we have . Hence , as claimed by the theorem.

Theorem 10. With the same notation as in the proof of the above theorem, the maximal divisor of all minors of is for .

Proof. First, we take care of the minors of , which is as , where . Assume that is a common divisor of , . Then where is minors of and . Then we can get that is a unit of . It follows that the maximal divisor of all minors of is .
Now let . Suppose that the result is correct for minors. We investigate this result for minors. Let be all the minors and all the submatrix of , where and . Then, by the Laplace expansion, But the common divisor of is a unit of , and the common divisor of is , so the common divisor of is . The theorem follows by induction on .

Besides, we can also make some improvements for the above theorem, which can be seen as the Serre problem generalized to elementary divisor rings.

Theorem 11. Let be an EDR and of full row rank. Then, for any maximal common divisor of all minors of , there exist and such that with and is ZLP.

Proof. Assume and are two maximal common divisors of all minors of . By Lemma 4, , which means there exist such that and . Note that as is of full row rank. Then and are not zero-divisors. Also, any two maximal common divisors of all minors of are associates of each other; that is, there exists a unit in such that .
By Theorem 9, there exist and such that , where is a maximal common divisor of all minors of , and is ZLP. Set . Then . Thus the theorem is proved.

Remark 12. The above two theorems are different from each other, as in Theorem 11   is beforehand given, but in Theorem 9   is an arbitrary one.
By now, we proved that Lin-Bose problem over an elementary divisor ring is correct. Next we deal with the Serre problem.

Theorem 13. Let be an elementary divisor ring and with . Then can be completed to a square matrix whose determinant is a maximal common divisor of all minors of . Furthermore, the matrix may be completed to a matrix in .

Proof. From Theorem 8, we have that , where , , and . Set . Then we have This implies that Furthermore, 1 is a maximal common divisor of all minors of . From above argument, the matrix may be chosen to be itself completable to a matrix in ().

In this theorem is a particular maximal common divisor. When is an arbitrary maximal common divisor, this theorem is also correct.

Theorem 14. Let be an EDR, let be of full row rank, and let be any maximal divisor of all minors of . Then can be completed to a square matrix whose determinant is . Furthermore, the matrix may be chosen to be itself completable to matrix in .

Proof. From Lemma 4, any two maximal common divisors of all minors of are associates. From Theorem 8, there exist and such that , where ; every is a divisor of , and . By Lemma 7, is a maximal common divisor of all minors of . Assume ; is a unit in . Set . Then , where and . Setting , we obtain the result.

Theorem 15. Let be an EDR and let be of full row rank. Assume , and where for . Then and for all .

Proof. By Theorem 11, there exist and such that . Then . For , we also have , and , where and . It follows that and .
Now for , there exist and such that . Then . So . So and all divide for .
We prove the theorem by induction on . If , it is obvious. Let . Suppose that the result is correct for ; we investigate this result for . By the definition of EDR and the assumption, we may set where . By the above, ; it follows that Hence Since , we conclude that So there exist such that That is, ThusNow, assume that . Then , and . But Therefore, we have , contradicting our assumption. So , and our theorem is proved.