Some properties of primitive matrices over Bezout B-domain

Some properties of primitive matrices over Bezout B-domain

The properties of primitive matrices (matrices for which the greatest common divisor of the minors of maximal order is equal to 1) over Bezout B - domain, i.e. commutative domain finitely generated principal ideal in which for all a,b,c with (a,b,c) = 1,c 6= 0, there exists element r ∈ R, such t...

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Date:2005
Main Author: Shchedryk, V.P.
Format: Article
Language:English
Published: Інститут прикладної математики і механіки НАН України 2005
Series:Algebra and Discrete Mathematics
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/156626
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Cite this:Some properties of primitive matrices over Bezout B-domain / V.P. Shchedryk // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 46–57. — Бібліогр.: 14 назв. — англ.

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spelling irk-123456789-1566262019-06-19T01:29:31Z Some properties of primitive matrices over Bezout B-domain Shchedryk, V.P. The properties of primitive matrices (matrices for which the greatest common divisor of the minors of maximal order is equal to 1) over Bezout B - domain, i.e. commutative domain finitely generated principal ideal in which for all a,b,c with (a,b,c) = 1,c 6= 0, there exists element r ∈ R, such that (a+rb,c) = 1 is investigated. The results obtained enable to describe invariants transforming matrices, i.e. matrices which reduce the given matrix to its canonical diagonal form. 2005 Article Some properties of primitive matrices over Bezout B-domain / V.P. Shchedryk // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 46–57. — Бібліогр.: 14 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 15A21. http://dspace.nbuv.gov.ua/handle/123456789/156626 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The properties of primitive matrices (matrices for which the greatest common divisor of the minors of maximal order is equal to 1) over Bezout B - domain, i.e. commutative domain finitely generated principal ideal in which for all a,b,c with (a,b,c) = 1,c 6= 0, there exists element r ∈ R, such that (a+rb,c) = 1 is investigated. The results obtained enable to describe invariants transforming matrices, i.e. matrices which reduce the given matrix to its canonical diagonal form.
format Article
author Shchedryk, V.P.
spellingShingle Shchedryk, V.P.
Some properties of primitive matrices over Bezout B-domain
Algebra and Discrete Mathematics
author_facet Shchedryk, V.P.
author_sort Shchedryk, V.P.
title Some properties of primitive matrices over Bezout B-domain
title_short Some properties of primitive matrices over Bezout B-domain
title_full Some properties of primitive matrices over Bezout B-domain
title_fullStr Some properties of primitive matrices over Bezout B-domain
title_full_unstemmed Some properties of primitive matrices over Bezout B-domain
title_sort some properties of primitive matrices over bezout b-domain
publisher Інститут прикладної математики і механіки НАН України
publishDate 2005
url http://dspace.nbuv.gov.ua/handle/123456789/156626
citation_txt Some properties of primitive matrices over Bezout B-domain / V.P. Shchedryk // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 46–57. — Бібліогр.: 14 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT shchedrykvp somepropertiesofprimitivematricesoverbezoutbdomain
first_indexed 2025-07-14T09:00:55Z
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2005). pp. 46 – 57 c© Journal “Algebra and Discrete Mathematics” Some properties of primitive matrices over Bezout B-domain V. P. Shchedryk Communicated by M.Ya. Komarnytskyj Abstract. The properties of primitive matrices (matrices for which the greatest common divisor of the minors of maximal order is equal to 1) over Bezout B - domain, i.e. commutative domain finitely generated principal ideal in which for all a, b, c with (a, b, c) = 1, c 6= 0, there exists element r ∈ R, such that (a+rb, c) = 1 is investigated. The results obtained enable to describe invariants transforming matrices, i.e. matrices which reduce the given matrix to its canonical diagonal form. The notation of elementary divisor ring, as rings over which every ma- trix admits diagonal reduction were introduced by I. Kaplansky in 1949 [1]. Such concept has appeared rather effective at the decision of many tasks in different areas of modern algebra. The whole direction in the theory of rings was formed in which the properties of rings of elementary divisor are studied, new classes of rings were described which possess the property of a diagonal reduction [2-5]. With current of time the interest to such rings has not died away – a lot of publications regularly occur in mathematical journals [6-9]. One of examples of a class elementary divisor ring are the Bezout B−rings, i.e. commutative domain of finitely generated principal ideal in which for all a, b, c with (a, b, c) = 1, c 6= 0, there exists an element r ∈ R, such that (a+ rb, c) = 1 [10]. This paper is devoted to study of Bezout B−domain from the point of view of re- search of properties of their elements, and also to describe the invariants of transformable matrices, i.e. invertible matrices which the given matrix reduces to its canonical diagonal form. 2000 Mathematics Subject Classification: 15A21. Key words and phrases: elementary divisor ring, Bezout B−domain, canonical diagonal form, transformable matrices, invariants, primitive matrices. Jo u rn al A lg eb ra D is cr et e M at h .V. P. Shchedryk 47 Let R be Bezout B−domain. A matrix is called primitive if the great- est common divisor of minors of the maximal order is equal to 1. In the first part of this paper the properties of primitive rows and columns are studied. Property 1. If (a1, . . . , an) = 1, an 6= 0, n ≥ 3, then there are elements u2, . . . , un−1, such that (a1 + u2a2 + . . .+ un−1an−1, an) = 1. Proof. Let (a2, . . . , an−1) = γ. Then there are elements v2, . . . , vn−1 , such that v2a2 + . . .+ vn−1an−1 = γ. Since (a1, γ, an) = 1, there is an element r ∈ R, for which (a1 + rγ, an) = 1. Thus (a1 + (rv2)a2 + . . .+ (rvn−1)an−1, an) = 1. Property 2. If (a1, . . . , an) = 1, an 6= 0, n ≥ 3, then there are invertible matrices of the form ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ a1 v1 v2 . . . vn−2 vn−1 a2 1 0 . . . 0 0 a3 0 1 0 0 . . . . . . . . . . . . . . . . . . an−1 0 0 1 0 an 0 0 . . . 0 vn ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ = V, ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ un 0 . . . 0 0 un−1 0 1 0 0 un−2 . . . . . . . . . . . . . . . . . . 0 0 1 0 u2 0 0 . . . 0 1 u1 an an−1 . . . a3 a2 a1 ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ = U. Proof. First we shall show, that the elements v1, . . . , vn can be chosen such that the matrix V will be invertible. By property 1 there are ele- ments v1, . . . , vn−2 , such that (a1 − v1a2 − . . .− vn−2an−1, an) = 1. Since detV = vnγn−1−vn−1an, where γn−1 = a1−v1a2− . . .−vn−2an−1, and taking into consideration (γn−1, a1) = 1 we can choose elements vn−1, vn so, that detV = 1. It is similarly shown, that there are u1, . . . , un for which detU = 1. Jo u rn al A lg eb ra D is cr et e M at h .48 Some properties of primitive matrices over... Since R is finitely generated principal ideal domain then for all finitely set of relatively prime elements a1, . . . , an, n ≥ 2, there are elements u1, . . . , un, such that u1a1 + . . .+ unan = 1. (1) Write the elements u1, . . . , un as ∥ ∥u1 . . . un ∥ ∥ . We shall say that el- ements of the row ∥ ∥u1 . . . un ∥ ∥ satisfies equation (1). The following statement suggests a method of finding of such all rows with elements which satisfy the equation (1). Property 3. Let (a1, . . . , an) = 1, n ≥ 2, and A be any invertible matrix for which ∥ ∥a1 . . . an ∥ ∥T is its first column. The set U = { ∥ ∥1 x2 . . . xn ∥ ∥A−1|xi ∈ R, i = 2, . . . , n} consist of all rows with elements which satisfy equation (1). Proof. Let ∥ ∥v1 . . . vn ∥ ∥ ∈ U, i.e. ∥ ∥v1 . . . vn ∥ ∥ = ∥ ∥1 x2 . . . xn ∥ ∥A−1, where xi ∈ R, i = 2, . . . , n. Then ∥ ∥v1 . . . vn ∥ ∥ ∥ ∥a1 . . . an ∥ ∥T = ∥ ∥1 x2 . . . xn ∥ ∥A−1 ∥ ∥a1 . . . an ∥ ∥T = = ∥ ∥1 x2 . . . xn ∥ ∥ ∥ ∥1 0 . . . 0 ∥ ∥T = 1. This means that elements of all rows from U satisfy equation (1). Let elements of the row ∥ ∥u1 . . . un ∥ ∥ satisfy equation (1) and A−1 = ‖bij‖ n 1 . Consider the matrix ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ u1 u2 . . . un b21 b22 . . . b2n . . . . . . . . . . . . bn1 bn2 . . . bnn ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ = U. Then UA = ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ 1 x2 . . . xn 0 1 . . . 0 ... . . . 0 0 . . . 1 ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ . It follows from this that ∥ ∥u1 . . . un ∥ ∥ ∈ U. This concludes the proof of our statement. Jo u rn al A lg eb ra D is cr et e M at h .V. P. Shchedryk 49 Property 4. If (a1, . . . , an) = 1, n ≥ 2, and ε1|ε2| . . . |εk, εi 6= 0, i = 1, . . . , k, 1 ≤ k < n, then (a1εk, a2εk−1, . . . , akε1, ak+1, . . . , an) = (εk, a2εk−1, . . . , akε1, ak+1, . . . , an) . Proof. Denote (a1εk, a2εk−1, . . . , akε1, ak+1, . . . , an) = δk. In order to prove this statement it suffices to show that δk|εk. In the case where k = 1 we have δ1 = (a1ε1, a2, . . . , an) = (a1ε1, (a2, . . . , an)) = (ε1, (a2, . . . , an)) . So δ1|ε1. Hence the results holds for k = 1. Let k ≥ 2 and suppose that the result is established for m < k. Then δk = (a1εk, a2εk−1, . . . , akε1, ak+1, . . . , an) = = ( ε1 ( a1 εk ε1 , . . . , ak−1 ε2 ε1 , ak ) , ak+1, . . . , an ) . Since ε2 ε1 | ε3 ε1 | . . . | εk ε1 we have by the induction hypothesis ( a1 εk ε1 , . . . , ak−1 ε2 ε1 , ak, ak+1, . . . , an ) = d1| εk ε1 . Therefore δk = d1  ε1 ( a1 εk ε1 , . . . , ak−1 ε2 ε1 , ak ) d1 , ( ak+1 d1 , . . . , an d1 )   = = d1 ( ε1, ( ak+1 d1 , . . . , an d1 )) = d1d2, where d2|ε1. Thus δk = d1d2| εk ε1 ε1 = εk. Property 5. Let (a, b, ϕ) = (a1, b1, ϕ) = 1, aba1b1ϕ 6= 0. If ab1 ≡ a1b(modϕ), then (ax+ b, ϕ) = (a1x+ b1, ϕ) for all x ∈ R. Jo u rn al A lg eb ra D is cr et e M at h .50 Some properties of primitive matrices over... Proof. Set (a, ϕ) = α. Then (a, b) = 1. As ab1 − a1b = ϕt, we have α|a1b. By the property 4 α|a1. Hence, α|(a1, ϕ) = α1. From similar reasons α1|(a, ϕ) = α. Hence α = α1. As (a1(ax+ b), ϕ) = (a1ax+ a1b, ϕ) = (a1ax+ (a1b+ ϕt), ϕ) = = (a1ax+ a1b, ϕ) = (a(a1x+ b1), ϕ), so (a1 α (ax+ b), ϕ α ) = ( a α (a1x+ b1), ϕ α ) . Therefore ( ax+ b, ϕ α ) = ( a1x+ b1, ϕ α ) = δ. Since α|a and α|a1, and also (α, b) = (α, b1) = 1 then for all elements x ∈ R the equality (ax+ b, α) = (a1x+ b1, α) = 1 holds. Hence δ = ( ax+ b, ϕ α ) = ( ax+ b, ϕ α α ) = (ax+ b, ϕ) . Similarly δ = (a1x+ b1, ϕ) . Property 6. Let (a1, . . . , an) = 1, n ≥ 2, and ψ ∈ R be any fixed nonzero element, which is not unit. Then there are elements u1, . . . , un, which satisfy the following conditions simultaneously: a) u1a1 + . . .+ unan = 1; b) (u1, . . . , ui) = 1, for any fixed 2 ≤ i ≤ n; c) (ui, ψ) = 1, for any fixed 2 ≤ i ≤ n. Proof. Consider the invertible matrixA with first column ∥ ∥a1 . . . an ∥ ∥T . Let’s show, that matrix A can be chosen in such a way that the elements of the matrix A−1 = ‖bij‖ n 1 satisfy b3i = . . . = bni = 0. Indeed, let A1 be any invertible matrix with first column ∥ ∥a1 . . . an ∥ ∥T , A−1 = ∥ ∥b̄ij ∥ ∥n 1 and among elements b̄2i, . . . , b̄ni there is at least one not zero. Then there is such a matrix D ∈ GLn−1(R), that D ∥ ∥b2i . . . bni ∥ ∥T = ∥ ∥γ 0 . . . 0 ∥ ∥T . Thus, the matrix ( (1 ⊕D)A−1 1 ) −1 = A1(1 ⊕D−1) = A Jo u rn al A lg eb ra D is cr et e M at h .V. P. Shchedryk 51 will be found. Let the matrix consisting of the first i columns of the matrix A−1 has the form ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ b11 . . . b1,i−1 b1i b21 . . . b2,i−1 γ b31 . . . b3,i−1 0 . . . . . . . . . . . . bi1 . . . bi,i−1 0 —————————— bi+1,1 . . . bi+1,i−1 0 . . . . . . . . . . . . bn1 . . . bn,i−1 0 ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ = ∥ ∥ ∥ ∥ M N ∥ ∥ ∥ ∥ . By property 3 every set of elements u1, . . . , un satisfying condition a) can be presented as follows: ∥ ∥u1 . . . un ∥ ∥ = ∥ ∥1 x2 . . . xn ∥ ∥A−1, where xi ∈ R, i = 2, . . . , n. In order that the our statement be valid it is sufficient, that there are elements x2 . . . xn, such that ∥ ∥1 x2 . . . xn ∥ ∥ ∥ ∥ ∥ ∥ M N ∥ ∥ ∥ ∥ = ∥ ∥q1 . . . qi ∥ ∥ , where (q1, . . . , qi) = (qi, ψ) = 1. Let γ = 0. Since the matrix ∥ ∥ ∥ ∥ M N ∥ ∥ ∥ ∥ is primitive, we conclude that b1i ∈ U(R). Therefore b11, . . . , b1i will be found elements. Let γ 6= 0 and btj 6= 0, i+1 ≤ t ≤ n, 1 ≤ j ≤ i−1. As (b1i, γ) = 1 then (b1i, γ, ψbtj) = 1. Therefore there is element l, such that (b1i +γl, ψbtj) = 1. This equality implies i) (d1i, ψ) = 1; (2) ii) (d1i, btj) = 1, where d1i = b1i + γl 6= 0. Then (d1j , btj , d1i) = 1, where d1j = b1j + b2jl. Therefore, there is m, such that (d1j +btjm, d1i) = 1. Taking into account equality (2), we are convinced, that elements of the first row of the matrix     ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ 1 l 0 . . . 0 m 0 1 0 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 1 ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ⊕ En−t     ∥ ∥ ∥ ∥ M N ∥ ∥ ∥ ∥ , Jo u rn al A lg eb ra D is cr et e M at h .52 Some properties of primitive matrices over... where En−t is identity (n − t) × (n − t) matrix, will satisfy to all the requirements of our statement. If N = 0, or i = n (in this case matrix N is empty) it follows from invertibility of a matrix A1, that M ∈ GLi(R). Therefore (b1i, γ) = 1, so (b1i, γ, ψ) = 1. As well as in the previous cases there is r, that (b1i + γr, ψ) = 1. Hence, the elements of the first row of the matrix (∥ ∥ ∥ ∥ 1 r 0 1 ∥ ∥ ∥ ∥ ⊕ Ei−2 ) M also will be found elements. The statement is proved. By the theorem 5.2 of [1] R is elementary divisor domain. Therefore for every nonsingular n× n matrix A over R exist invertible matrices P and Q (further we shall call them as transformable matrices), such that PAQ = diag(ϕ1, . . . , ϕn) = Φ, ϕi|ϕi+1, i = 1, . . . , n− 1. (3) Denote PA the set of invertible matrices P , which satisfy equality (3). In a final part of this paper the properties of set of transformable matrices PA will be studied. It was shown in papers [11-13], that PA = GΦP , where P be any fixed matrix from equality (3), and GΦ is multiplicative group, which consists of all invertible matrices of the form ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ h11 h12 . . . h1,n−1 h1n ϕ2 ϕ1 h21 h22 . . . h2,n−1 h2n . . . . . . . . . . . . . . . ϕn ϕ1 hn1 ϕn ϕ2 h22 . . . ϕn ϕn−1 hn,n−1 hnn ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ . In [12,14] it is proved that the group GΦ and the set of left trans- formable matrices PA play the main role in the description of the asso- ciated matrices, which have the given canonical diagonal form Φ. Proposition ([12, 14]). Let A = P−1 A ΦQ−1 A , B = P−1 B ΦQ−1 B . The follow- ing are equivalent: a) A and B are right associates (B = AU, U ∈ GLn(R)); b) PB = HPA, where H ∈ GΦ; c) PB = PA. We apply the obtained results to describe the properties of trans- formable matrices. Denote Jo u rn al A lg eb ra D is cr et e M at h .V. P. Shchedryk 53 Φ1 = En,Φi = diag   ϕi ϕ1 , . . . , ϕi ϕi−1 , 1, . . . , 1 ︸ ︷︷ ︸ n−i+1   , i = 2, . . . , n. Definition 1. Let ∥ ∥a1 . . . an ∥ ∥T be primitive column and Φi ∥ ∥a1 . . . an ∥ ∥T ∼ ∥ ∥δi 0 . . . 0 ∥ ∥T , i = 1, . . . , n. The column ∥ ∥δ1 . . . δn ∥ ∥T is called Φ− rod of the column ∥ ∥a1 . . . an ∥ ∥T . Theorem 1. If H ∈ GΦ, then Φ-rods of columns ∥ ∥a1 . . . an ∥ ∥T , H ∥ ∥a1 . . . an ∥ ∥T coincide. Proof. Since j column of the matrix H has the form ∥ ∥ ∥h1j . . . hjj ϕj+1 ϕj hj+1,j . . . ϕn ϕj hnj ∥ ∥ ∥ T , 1 ≤ j ≤ n− 1, then Φihj = = ∥ ∥ ∥ ϕi ϕ1 h1j . . . ϕi ϕj−1 hj−1,j ϕi ϕj hjj ϕi ϕj hj+1,j . . . ϕi ϕj hij ϕi+1 ϕj hi+1,j . . . ϕn ϕj hnj ∥ ∥ ∥ T = ϕi ϕj ∥ ∥ ∥ ϕj ϕ1 h1j . . . ϕj ϕj−1 hj−1,j hjj . . . hij ϕi+1 ϕi hi+1,j . . . ϕn ϕi hnj ∥ ∥ ∥ T . Hence, ϕi ϕj |Φihj , i = 2, . . . , n, j = 1, . . . , n − 1, i > j. It means that the equalities ΦiH = KiΦi, (4) i = 2, . . . , n, holds. As all the matrices Φi are nonsingular, and the matrix H is invertible, then from equality (4) follows Ki ∈ GLn(R). Therefore ΦiH ∥ ∥a1 . . . an ∥ ∥T = KiΦi ∥ ∥a1 . . . an ∥ ∥T ∼ ∼ Φi ∥ ∥a1 . . . an ∥ ∥T ∼ ∥ ∥δi 0 . . . 0 ∥ ∥T , i = 2, . . . , n. It remains to note that δ1 = 1 which concludes the proof of the theorem. Jo u rn al A lg eb ra D is cr et e M at h .54 Some properties of primitive matrices over... Theorem 2. Let ∥ ∥δ1 . . . δn ∥ ∥T be Φ-rod of the primitive column ∥ ∥a1 . . . an ∥ ∥T . Then there is a matrix H ∈ GΦ for which H ∥ ∥a1 . . . an ∥ ∥T = ∥ ∥b δ2 . . . δn ∥ ∥T . Proof. Property 6 implies that there are elements u1, . . . , un, such that ϕn ϕ1 u1a1 + . . .+ ϕn ϕn−1 un−1an−1 + unan = δn, where ( un, ϕn ϕ1 ) = 1. Since ϕn ϕn−1 | ϕn ϕn−2 | . . . |ϕn ϕ1 , taking in account the prop- erty 4 ( ϕn ϕ1 u1, . . . , ϕn ϕn−1 un−1, un ) = ( ϕn ϕ1 , ϕn ϕ2 u2, . . . , ϕn ϕn−1 un−1, un ) = = ( ϕn ϕ2 u2, . . . , ϕn ϕn−1 un−1, ( un, ϕn ϕ1 )) = 1. By property 2 we shall complete a primitive row ∥ ∥ ∥ ϕn ϕ1 u1 . . . ϕn ϕn−1 un−1 un ∥ ∥ ∥ to an invertible matrix Hn in which this row will be last, and other el- ements of this matrix, which lies under the main diagonal will be zero. Then Hn ∈ GΦ and Hn ∥ ∥a1 . . . an ∥ ∥T = ∥ ∥b1 . . . bn−1 δn ∥ ∥T . By theorem 1 this column will have again Φ-rod ∥ ∥δ1 . . . δn ∥ ∥T . There- fore Φn−1Hn ∥ ∥a1 . . . an ∥ ∥T ∼ ∥ ∥δn−1 0 . . . 0 ∥ ∥T . Hence, there are elements v1, . . . , vn, such that ϕn−1 ϕ1 v1b1 + . . .+ ϕn−1 ϕn−2 vn−2bn−2 + vn−1bn−1 + vnδn = δn−1. Moreover, as it follows from property 6, these elements can be chosen in such a manner that (v1, . . . , vn−1) = 1 and ( vn−1, ϕn−1 ϕ1 ) = 1. Thus we have ( ϕn−1 ϕ1 v1, . . . , ϕn−1 ϕn−2 vn−2, vn−1 ) = 1. Jo u rn al A lg eb ra D is cr et e M at h .V. P. Shchedryk 55 It means, that in the group GΦ there is a matrix Hn−1 with the following two last rows: ∥ ∥ ∥ ∥ ϕn−1 ϕ1 v1 . . . ϕn−1 ϕn−2 vn−2 vn−1 vn 0 . . . 0 0 1 ∥ ∥ ∥ ∥ . Consequently, Hn−1Hn ∥ ∥a1 . . . an ∥ ∥T = ∥ ∥d1 . . . dn−2 δn−1 δn ∥ ∥T . Continuing the described process, on (n-1) step we shall receive the matrix H = H2 · · ·Hn ∈ GΦ, such that HA = ∥ ∥b δ2 . . . δn ∥ ∥T . The theorem is proved. Denote ∆i = ( ϕi ϕi−1 , ai δi−1 , . . . , an δi−1 ) , i = 2, . . . , n. (5) Theorem 3. Let ∥ ∥δ1 . . . δn ∥ ∥T be Φ-rod of the primitive column ∥ ∥a1 . . . an ∥ ∥T . Then the elements δi satisfy the following conditions: a) δi = ∆2 · · ·∆i, i = 2, . . . , n; b) δi| ϕi ϕ1 , i = 2, . . . , n. Proof. Since δ1 = 1, we obtain from property 4 δ2 = ( ϕ2 ϕ1 a1, a2, . . . , an ) = ∆2, δ3 = ( ϕ3 ϕ1 a1, ϕ3 ϕ2 a2, a3, . . . , an ) = ( ϕ3 ϕ2 ( ϕ2 ϕ1 , a2 ) , a3, . . . , an ) = = δ2   ϕ3 ϕ2 ( ϕ2 ϕ1 , a2 ) δ2 , a3 δ2 , . . . , an δ2   = ∆2 ( ϕ3 ϕ2 , a3 δ2 , . . . , an δ2 ) = ∆2∆3. Having continued on analogy our reasons, we obtain δi = ∆2 · · ·∆i, i = 2, . . . , n. By (5) ∆i| ϕi ϕi−1 , i = 2, . . . , n. Hence δi = ∆2∆3 · · ·∆i−1∆i| ϕ2 ϕ1 ϕ3 ϕ2 · · · ϕi−1 ϕi−2 ϕi ϕi−1 = ϕi ϕ1 , i = 2, . . . , n. Jo u rn al A lg eb ra D is cr et e M at h .56 Some properties of primitive matrices over... The following corollary follows from the theorems 2 and 3. Corollary 1. If ∥ ∥δ1 . . . δn ∥ ∥T be Φ-rod of the primitive column ∥ ∥a1 . . . an ∥ ∥T , then there is a matrix H ∈ GΦ, such that H ∥ ∥a1 . . . an ∥ ∥T = ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ b ∆2 ∆2∆3 . . . ∆2∆3 · · ·∆n ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ , where ∆i| ϕi ϕi−1 , i = 2, . . . , n. Definition 2. Let P ∈ GLn(R) and p̄1, . . . , p̄n be its columns, ∥ ∥δi1 . . . δin ∥ ∥T is Φ− rod of column p̄i, i = 1, . . . , n. The matrix ‖δij‖ n 1 is called Φ − rod of the matrix P . Theorem 4. Φ-rods of matrices from PA coincide. Proof. Let P1 be any matrix from PA. Since PA = GΦP then there exists a matrix H ∈ GΦ, such that P1 = HP. According to the theorem 1 Φ- rods correspond columns of matrices P and P1 coincide. Therefore will be Φ-rods of these matrices coincide. Since all matrices of the set PA have identical Φ-rods, it is possible to speak about Φ-rod of set of transformable matrices PA, having identified it with Φ-rod of any matrix of this set. Using the theorem 2, we obtain. Corollary 2. Let ‖δij‖ n 1 be Φ-rod of the set PA. Then there is matrix Pi ∈ PA, which have i column of the form ∥ ∥∗ δi1 . . . δin ∥ ∥T , 1 ≤ i ≤ n. Corollary 3. If the matrices A and B have the canonical diagonal form Φ and are right associates, then Φ-rods of sets PA and PB coincide. Proof. By proposition PA = PB, so that Φ-rods of these sets coincide. References [1] Kaplansky I. Elementary divisor ring and modules, Trans. Amer. Math. Soc.Vol. 66.– 1949.– 464–491. [2] Gillman L., Henriksen M. Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. Vol. 82.– 1956.– 362–365. [3] Henriksen M. On a class of regular rings that are elementary divisor rings, Arch. Math. Vol. 42 . 1973 . 133–141 [4] Larsen M., Lewis W., Shores T. Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc.Vol. 187 . 1974 . 231–248 Jo u rn al A lg eb ra D is cr et e M at h .V. P. Shchedryk 57 [5] Menal P., Moncasi I. On regular rings with stable range 2, J. Pure Appl. 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P., The structure and properties of divisors of matrices over com- mutative domain elementary divisors ring, Mat. StudiiVol. 10:2. 1998. 115–120, in Ukrainian [13] Kazimirs’kij P. S., A solution to the problem of separating a regular factor from a matrix polynomial, Ukr. Mat. ZhVol. 32:4. 1980. 483–498, in Russian [14] Mel’nik O.M. On invariants of transforming matrices Methods for the investiga- tions of differential and integral operators, Collect. Sci. Works, Kiev. yr 1989 . 160-164, in Russian Contact information V. P. Shchedryk Department of Algebra Pidsryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine 3b Naukova Str. Lviv, 79060, UKRAINE Received by the editors: 11.05.2004 and final form in 08.05.2005.

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