Reducibility and irreducibility of monomial matrices over commutative rings

Reducibility and irreducibility of monomial matrices over commutative rings

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Бібліографічні деталі
Дата:2013
Автори: Bondarenko, V.M., Bortos, M.Yu., Dinis, R.F., Tylyshchak, A.A.
Формат: Стаття
Мова:English
Опубліковано: Інститут прикладної математики і механіки НАН України 2013
Назва видання:Algebra and Discrete Mathematics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/152344
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Цитувати:Reducibility and irreducibility of monomial matrices over commutative rings / V.M. Bondarenko, M.Yu. Bortos, R.F. Dinis, A.A. Tylyshchak // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 171–187. — Бібліогр.: 6 назв. — англ.

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spelling irk-123456789-1523442019-06-11T01:25:04Z Reducibility and irreducibility of monomial matrices over commutative rings Bondarenko, V.M. Bortos, M.Yu. Dinis, R.F. Tylyshchak, A.A. 2013 Article Reducibility and irreducibility of monomial matrices over commutative rings / V.M. Bondarenko, M.Yu. Bortos, R.F. Dinis, A.A. Tylyshchak // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 171–187. — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC:15B33, 15A30. http://dspace.nbuv.gov.ua/handle/123456789/152344 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
format Article
author Bondarenko, V.M.
Bortos, M.Yu.
Dinis, R.F.
Tylyshchak, A.A.
spellingShingle Bondarenko, V.M.
Bortos, M.Yu.
Dinis, R.F.
Tylyshchak, A.A.
Reducibility and irreducibility of monomial matrices over commutative rings
Algebra and Discrete Mathematics
author_facet Bondarenko, V.M.
Bortos, M.Yu.
Dinis, R.F.
Tylyshchak, A.A.
author_sort Bondarenko, V.M.
title Reducibility and irreducibility of monomial matrices over commutative rings
title_short Reducibility and irreducibility of monomial matrices over commutative rings
title_full Reducibility and irreducibility of monomial matrices over commutative rings
title_fullStr Reducibility and irreducibility of monomial matrices over commutative rings
title_full_unstemmed Reducibility and irreducibility of monomial matrices over commutative rings
title_sort reducibility and irreducibility of monomial matrices over commutative rings
publisher Інститут прикладної математики і механіки НАН України
publishDate 2013
url http://dspace.nbuv.gov.ua/handle/123456789/152344
citation_txt Reducibility and irreducibility of monomial matrices over commutative rings / V.M. Bondarenko, M.Yu. Bortos, R.F. Dinis, A.A. Tylyshchak // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 171–187. — Бібліогр.: 6 назв. — англ.
series Algebra and Discrete Mathematics
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AT tylyshchakaa reducibilityandirreducibilityofmonomialmatricesovercommutativerings
first_indexed 2025-07-13T02:52:12Z
last_indexed 2025-07-13T02:52:12Z
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fulltext Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 16 (2013). Number 2. pp. 171 – 187 c© Journal “Algebra and Discrete Mathematics” Reducibility and irreducibility of monomial matrices over commutative rings Vitaliy M. Bondarenko, Maria Yu. Bortos, Ruslana F. Dinis, Alexander A. Tylyshchak Communicated by V. V. Kirichenko Dedicated to the memory of Professor Petro M. Gudivok Abstract. Let R be a local ring with nonzero Jacobson radical. We study monomial matrices over R of the form   0 ... 0 t sn t s1 ... 0 0 ... . . . ... ... 0 ... t s n−1 0   , and give a criterion for such matrices to be reducible when n ≤ 6, s1 . . . , sn ∈ {0, 1} and the radical is a principal ideal with generator t. We also show that the criterion does not hold for n = 7. Introduction The problem of classifying, up to similarity, all the matrices over a commutative ring (which is not a field) is usually very difficult; in most cases it is “unsolvable” (wild), as in the case of the rings of residue classes [1]. Special cases of matrices of small orders were considered by many authors (see, e.g., [2]–[5]). In such situation, an important place is occupied by irreducible matrices over rings. Our paper is devoted to this subject. 2010 MSC: 15B33, 15A30. Key words and phrases: irreducible matrix, similarity, local ring, Jacobson radical. 172 Reducibility and irreducibility of monomial matrices Throughout the paper R denotes a commutative ring with identity, which is not a field, and R∗ the group of its invertible elements. We say that an n × n matrix M over R is reducible over R, or simply reducible, if it is similar (over R) to a matrix N = ( A1 B 0 A2 ) , where Ai is an ni × ni matrix over R; i = 1, 2, n1, n2 > 0 (i. e. there exists an invertible matrix X over R such that X−1MX = N). Otherwise we say that M is irreducible over R, or simply irreducible. We consider the question: when is an n × n matrix over R of the form M(t, s1, . . . , sn) =       0 . . . 0 tsn ts1 . . . 0 0 ... . . . ... ... 0 . . . tsn−1 0       (1) with t ∈ R irreducible? The answer to this question is only known in some cases. Obviously, M(t, s1, . . . , sn) is reducible if ts1 = . . . = tsn with n > 1 (in particular, t ∈ {0, 1} or s1 = . . . = sn). If R is local and its radical is a principle ideal with generator t, then M(t, 0, . . . , 0, 1) is irreducible since its characteristic polynomial xn +(−1)n+1t is irreducible; and M(t, 0, 1, . . . , 1) is irreducible (probed by Gudivok and Tylyshchak [6]). 1. Reducible matrices M(t, s1, . . . , sn) Z[λ] denotes the ring of polynomials of the variable λ over the ring Z of integer numbers. Its field of fractions is denoted by F . By a basis of a vector space we mean an ordered basis. As usual, n denotes a natural number. Proposition 1. Let s1, . . . , sn be natural numbers such that s = ∑n i=1 si and n are not coprime. Then for any common divisors d > 1 of s and n, the matrix M = M(λ, s1, . . . , sn) over Z[λ] is similar (over Z[λ]) to a matrix of the form N = ( A D 0 B ) , where A is an n d × n d matrix. Bondarenko, Bortos, Dinis, Tylyshchak 173 Proof. Let s = dk, n = dm. To illustrate the idea of the proof, we consider the following special case: n s s1, . . . , sn d m k 15 9 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1 3 5 3 Let e = {e1, e2, . . . , e15} be the standard basis of the vector space F 15 and let ϕ be the linear operator on this vector space determined (in the basis e) by the matrix M(λ, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1), which has, by definition, the form                               0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 0                               . Then ϕ(e1) = e2, ϕ(e2) = e3, ϕ(e3) = e4, ϕ(e4) = e5, ϕ(e5) = e6, ϕ(e6) = e7, ϕ(e7) = λe8, ϕ(e8) = λe9, ϕ(e9) = λe10, ϕ(e10) = λe11, ϕ(e11) = λe12, ϕ(e12) = λe13, ϕ(e13) = λe14, ϕ(e14) = λe15, ϕ(e15) = λe1. (2) We can write these equalities in the form of the following diagram: 174 Reducibility and irreducibility of monomial matrices e5 tt e4 oo e6 ~~ e3 ee e7 λ �� e2 ZZ e8 λ �� e1 WW e9 λ �� e15 λ GG e10 λ e14 λ DD e11 λ ** e12 λ // e13 λ 99 where ei // ej and ei λ // ej mean respectively that ϕ(ei) = ej and ϕ(ei) = λej . Obviously, ϕ15(e1) = λ9e1. Put b′ 1 = λ6e1 + λ3ϕ5(e1) + ϕ10(e1) = λ6e1 + λ3e6 + λ4e11. (3) On the diagram e5 tt e4 oo e6 ~~ e3 ee e7 λ �� e2 ZZ e8 λ �� b′ 1 // �� ZZ e1 VV e9 λ �� e15 λ HH e10 λ e14 λ DD e11 λ ** e12 λ // e13 λ 99 Bondarenko, Bortos, Dinis, Tylyshchak 175 b′ 1 // ei indicate those ei which appeared in (3). Then ϕ5(b′ 1) = ϕ5(λ6e1 + λ3ϕ5(e1) + ϕ10(e1)) = λ6ϕ5(e1) + λ3ϕ10(e1) + ϕ15(e1) = λ6ϕ5(e1) + λ3ϕ10(e1) + λ9e1 = λ9e1 + λ6ϕ5(e1) + λ3ϕ10(e1) = λ3(λ6e1 + λ3ϕ5(e1) + ϕ10(e1)) = λ3b′ 1. Let us define b′ 2, . . . , b′ 5 by recursion b′ 2 = ϕ(b′ 1), b′ 3 = ϕ(b′ 2), b′ 4 = ϕ(b′ 3), b′ 5 = ϕ(b′ 4). (4) Clearly b′ 1 = λ6e1 + λ3e6 + λ4e11, b′ 2 = ϕ(b′ 1) = λ6e2 + λ3e7 + λ5e12, b′ 3 = ϕ(b′ 2) = λ6e3 + λ4e8 + λ6e13, b′ 4 = ϕ(b′ 3) = λ6e4 + λ5e9 + λ7e14, b′ 5 = ϕ(b′ 4) = λ6e5 + λ6e10 + λ8e15, and ϕ(b′ 5) = λ6e6 + λ7e12 + λ9e1 = λ9e1 + λ6e6 + λ7e12. Then ϕ(b′ 5) = ϕ5(b′ 1) = λ3b′ 1. From (2)–(4) it follows that b′ j = λα1j ej + λα2j e5+j + λα3j e10+j for some integer αij ≥ 0; here i = 1, 2, 3, j = 1, . . . , 5. Put αj = min i {αij}. Then α1 = 3, α2 = 3, α3 = 4, α4 = 5, α5 = 6, whence αj ≥ αj−1 (j > 1), α1 + 3 = α5. Let βij = αij − αj (i = 1, 2, 3, j = 1, . . . , 5). Then βij ≥ 0, and obviously that for any j there is 1 ≤ µj ≤ 3 such that βµj j = 0. Put bj = d ∑ i=1 λβij e(i−1)5+j or more detail b1 = λ3e1 + e6 + λe11, b2 = λ3e2 + e7 + λ2e12, b3 = λ2e3 + e8 + λ2e13, b4 = λe4 + e9 + λ2e14, b5 = e5 + e10 + λ2e15. (5) 176 Reducibility and irreducibility of monomial matrices Then λα1b1 = λ3b1 = λ6e1 + λ3e6 + λ4e11 = b′ 1, λα2b2 = λ3b2 = λ6e2 + λ3e7 + λ5e12 = b′ 2, λα3b3 = λ4b3 = λ6e3 + λ4e8 + λ6e13 = b′ 3, λα4b4 = λ5b4 = λ6e4 + λ5e9 + λ7e14 = b′ 4, λα5b5 = λ6b5 = λ6e5 + λ6e10 + λ8e15 = b′ 5. It follows from (4) that ϕ(bj−1) = λαj−αj−1bj (j = 2, . . . , 5) and ϕ(b5) = λ3+α1−α5b1 (since ϕ(b′ 5) = λ3b′ 1). Then ϕ(b1) = λ3−3b2 = b2, ϕ(b2) = λ4−3b2 = λb3, ϕ(b3) = λ5−4b3 = λb4, ϕ(b4) = λ6−5b4 = λb5, ϕ(b5) = λ3+3−6b1 = b1. (6) Denote by a the following basis of F 15: a = {e6, e7, e8, e9, e5, e1, e2, e3, e4, e10, e11, e12, e13, e14, e15}. The transition matrix from the basic e to the basis a is the (permuta- tion) matrix P =                               0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                               ∈ GL15(Z). From (5) it follows that b = {b1, b2, b3, b4, b5, e1, e2, e3, e4, e10, e11, e12, e13, e14, e15} Bondarenko, Bortos, Dinis, Tylyshchak 177 is a basis of F 15. The transition matrix from the basic a to the basis b is C =                               1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 λ3 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 λ3 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 λ2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 λ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 λ2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 λ2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 λ2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 λ2 0 0 0 0 0 0 0 0 0 1                               (belonging obviously to GL15(Z[λ])). Consider now the matrix S = PC ∈ GLn(Z[λ]), which is the transition matrix from the basic e to the basis b. It follows from (6) that S−1MS = ( A D 0 B ) (7) where A =        0 0 0 0 1 1 0 0 0 0 0 λ 0 0 0 0 0 λ 0 0 0 0 0 λ 0        and B is an 10 × 10 matrix over the ring Z[λ]. This completes the proof in our special case. Now we proceed to the general case. Recall that s = ∑n i=1 si = dk, n = dm. Let ϕ be the linear operator on the vector space F n determined by the matrix M in the standard basis e = {e1, e2, . . . , en}. Then ϕ(e1) = λs1e2, ϕ(e2) = λs2e3, . . . , ϕ(en−1) = λsn−1en, ϕ(en) = λsne1. Thus ϕ(ei) = { λsiei+1, i < n, λsie1, i = n. (8) 178 Reducibility and irreducibility of monomial matrices Obviously, ϕn(e1) = λ ∑n i=1 sie1 = λse1. Let b′ 1 = d ∑ i=1 λ(d−i)kϕ(i−1)m(e1). (9) Then ϕm(b′ 1) = ϕm ( d ∑ i=1 λ(d−i)kϕ(i−1)m(e1) ) = d ∑ i=1 λ(d−i)kϕim(e1) = d−1 ∑ i=1 λ(d−i)kϕim(e1) + ϕdm(e1) = ϕdm(e1) + d−1 ∑ i=1 λ(d−i)kϕim(e1) = ϕn(e1) + d ∑ i=2 λ(d−i+1)kϕ(i−1)m(e1) = = λse1 + d ∑ i=2 λ(d−i+1)kϕ(i−1)m(e1) = λdke1 + d ∑ i=2 λ(d−i+1)kϕ(i−1)m(e1) = d ∑ i=1 λ(d−i+1)kϕ(i−1)m(e1) = λk ( d ∑ i=1 λ(d−i)kϕ(i−1)m(e1) ) = λkb′ 1. Let us define b′ j by the recursion b′ j = ϕ(b′ j−1), (j = 2, . . . , m). (10) Then ϕ(b′ m) = ϕm(b′ 1) = λkb′ 1. From (8)–(10), it follows that b′ j = d ∑ i=1 λαij e(i−1)m+j for some αij ∈ Z (i = 1, . . . , d, j = 1, . . . , m). Moreover, b′ j = ϕ(b′ j−1) = d ∑ i=1 λαij−1ϕ(e(i−1)m+j−1) = = d ∑ i=1 λαij−1+s(i−1)m+j−1e(i−1)m+j (j = 2, . . . , m). Consequently, αij = αij−1 + s(i−1)m+j−1. Thus αij ≥ αi j−1 (i = 1, . . . , d, j = 2, . . . , m). Put αj = min i {αij} (j = 1, . . . , m). Then αj ≥ αj−1 (j > 1). Since d ∑ i=1 λαi1+ke(i−1)m+1 = λkb′ 1 = ϕ(b′ m) = d ∑ i=1 λαi mϕ(e(i−1)m+m) = Bondarenko, Bortos, Dinis, Tylyshchak 179 = d ∑ i=1 λαi mϕ(eim) = d−1 ∑ i=1 λαi mϕ(eim) + λαd mϕ(edm) = = d−1 ∑ i=1 λαi m+sim(eim+1) + λαd m+sne1 = λαd m+sne1 + d−1 ∑ i=1 λαi m+sim(eim+1) = = λαd m+sne1 + d ∑ i=2 λαi−1,m+s(i−1)m(e(i−1)m+1) we deduce that αi1 + k ≥ αi−1,m (i = 2, . . . , d). Moreover, α11 + k ≥ αd m. Thus α1 + k ≥ αm. Put βij = αij − αj (i = 1, . . . , d, j = 1, . . . , m). Then βij ≥ 0 for all i, j and, obviously, for any j there is 1 ≤ µj ≤ d such that βµj ,j = 0. Let bj = d ∑ i=1 λαij−αj e(i−1)m+j . Then λαj bj = d ∑ i=1 λαij e(i−1)m+j = b′ j , and it follows from (10) that ϕ(bj−1) = λαj−αj−1bj (j = 2, . . . , m). Since ϕ(b′ m) = λkb′ 1, we deduce that ϕ(bm) = λk+α1−αmb1. Let β(j − 1) = αj − αj−1, β(m) = k + α1 − αm. Clearly β(j) ≥ 0 (j = 1, . . . , m). Moreover, ϕ(bj−1) = λβ(j−1)bj (j = 2, . . . , m), ϕ(bm) = λβ(m)b1. Consider the vectors e(µ1−1)m+1, e(µ2−1)m+2, . . . , e(µm−1)m+m (belong- ing to the original basic e). They are distinct because their indices are not congruent modulo m. Therefore we can extend these vectors to a basis a = {e(µ1−1)m+1, e(µ2−1)m+2, . . . , e(µm−1)m+m, ei′ m+1 , . . . , ei′ n } of F n which is equal, up to a permutation, to the basic e = {e1, . . . , en}. The transition matrix from the basic a to a basis b = {b1, . . . , bm, ei′ m+1 , . . . , ei′ n } has the form C =            1 . . . 0 0 . . . 0 ... . . . ... ... . . . ... 0 . . . 1 0 . . . 0 λδm+1,1 . . . λδm+1,m 1 . . . 0 ... . . . ... ... . . . ... λδn1 . . . λδnm 0 . . . 1            180 Reducibility and irreducibility of monomial matrices where δij ≥ 0 (i = m+1, . . . , n, j = 1, . . . , m). Obviously, C ∈ GLn(Z[λ]) (as an matrix over Z[λ] with determinant 1). Let P ∈ GLn(Z) be a permutation matrix which is the transition matrix from the basic e to the basis a. Then the matrix S = PC is the transition matrix from the basic e to the basis b, and since ϕ(bj−1) = λβ(j−1)bj (j = 2, . . . , m), ϕ(bm) = λβ(m)b1 we deduce that S−1MS = ( A D 0 B ) , where A =         0 0 . . . 0 λβ(m) λβ(1) 0 . . . 0 0 0 λβ(2) . . . 0 0 ... ... . . . ... ... 0 0 . . . λβ(m−1) 0         , and B is a (n − m) × (n − m) matrix. Theorem 1. Let R be a commutative local ring, and let n > 0 and s1, . . . , sn ≥ 0 be integer numbers such that n and s = ∑n i=1 si are not coprime. Then for any common divisors d > 1 of n and s, and any t ∈ R the matrix M(t, s1, . . . , sn) =       0 . . . 0 tsn ts1 . . . 0 0 ... . . . ... ... 0 . . . tsn−1 0       over the ring R is reducible. The proof follows from Proposition 1 and the existence of a (unique) homomorphism of rings f : Z[λ] → R such that f(1) = 1, f(λ) = t. 2. Irreducible matrices M(t, s1, . . . , sn) Throughout this section all matrices are considered over a commutative local ring R with Jacobson radical Rad(R) = tR, t 6= 0, and their similarity are considered also over R. For a matrix M , we denote by M its reduction modulo the radical. As above n denotes a natural number. Bondarenko, Bortos, Dinis, Tylyshchak 181 Lemma 1. Let s1, . . . , sn ∈ {0, 1} with si = 0 for at least one i ∈ {1, . . . , n}. Then the matrix M = M(t, s1, . . . , sn) is not similar to ma- trices of the form M1 = ( tA D 0 B ) , M2 = ( A D 0 tB ) , where A is an r×r matrix and B is an (n−r)×(n−r) matrix (0 < r < n). Proof. Suppose that C−1MC = M1, where C = (cij)1≤i,j≤n ∈ GLn(R), or equivalently, MC = CM1, i. e.       0 . . . 0 tsn ts1 . . . 0 0 ... . . . ... ... 0 . . . tsn−1 0       C = C ( tA D 0 B ) . (11) For i, j ∈ {1, . . . , n}, the scalar equality (MC)ij = (CM1)ij is denoted by (11, ij). Put ci = (ci1, . . . , cir). We write the equalities (11, 1j), (11, 2j), . . . , (11, nj), where, in all cases, j runs from 1 to r, respectively in the form tsncn = tc1A, ts1c1 = tc2A, . . . , tsn−1cn−1 = tcnA. (12) Since Rad(R) is generated by t 6= 0, we have the following simple fact: if tsγ = tδ for some s ∈ {0, 1}, γ, δ ∈ R then either s = 0 and consequently γ = tδ, or s = 1 and consequently t(γ − δ) = 0; so, respectively, γ = 0 or γ = δ. If one put cn+1 = c1, then by this fact ci = 0 or ci = ci+1A for any i = 1, . . . , n. Because si = 0 for at least one i, we have that cl = 0 for some l ∈ {1, . . . , n}. By (12) we successively obtain c1 = . . . cl−1 = 0, cn = 0 and cl+1 = . . . cn−1 = 0. Thus the matrix C in not invertible modulo t and consequently is not invertible itself, a contradiction. The case C−1MC = M2 is considered analogously. Lemma 2. Let s1, . . . , sn ≥ 0 be integers with si 6= 0 for at least one i ∈ {1, . . . , n}. The matrix M = M(t, s1, . . . , sn) is not similar to a matrix N = ( A D 0 B ) , 182 Reducibility and irreducibility of monomial matrices where A is an r×r matrix and B is an (n−r)×(n−r) matrix (0 < r < n), A or B is invertible. The lemma follows at once from the nilpotency of M . Lemma 3. The matrix M(t, 0, 0, 0, 1, 1) is irreducible. Proof. Assume that the matrix M(t, 0, 0, 0, 1, 1) is reducible. Then for some matrix C ∈ GL5(R) we have M(t, 0, 0, 0, 1, 1)C = C ( A D 0 B ) . (13) where A is an s × s matrix and B is a (5 − s) × (5 − s) matrix (0 < s < 5). Since the matrix M(t, 0, 0, 0, 1, 1) is similar to its transpose, we can interchange the matrices A and B in (13), and therefore, without loss of generality, we can assume that s ≤ 5 − s, i. e. s = 1 or s = 2. In the case s = 1 either A ∈ tR or A ∈ R∗, and we have a contradiction with Lemmas 1 or 2, respectively. Let now s = 2 and let C = (cij)1≤i,j≤5, cp = (cp1, cp2) (p = 1, . . . , 5). Then from the equality (13) we obtain tc5 = c1A, c1 = c2A, c2 = c3A, c3 = c4A, tc4 = c5A. (14) By Lemmas 1 and 2 rank(A) 6= 0 and rank(A) 6= 2. Consequently rank(A) = 1. Since the matrix A is nilpotent, we can assume, without loss of generality, that A = ( tα 1 + tβ tγ tδ ) , where α, β, γ, δ ∈ R. Substituting A in (14), we get the following equalities (which for convenience are written in pairs):              (tc51, tc52) = (c11tα + c12tγ, c11 + c11tβ + c12tδ), (c11, c12) = (c21tα + c22tγ, c21 + c21tβ + c22tδ), (c21, c22) = (c31tα + c32tγ, c31 + c31tβ + c32tδ), (c31, c32) = (c41tα + c42tγ, c41 + c41tβ + c42tδ), (tc41, tc42) = (c51tα + c52tγ, c51 + c51tβ + c52tδ). (15) From the equalities (15) we obtain that c11 = c21 = c31 = 0 = c51. Further, c12 = 0 (since c12 = c21 + c21tβ + c22tδ and c21 = 0), and c22 = 0 (since c22 = c31 + c31tβ + c32tδ and c31 = 0). Bondarenko, Bortos, Dinis, Tylyshchak 183 Finally, since c41 = c51α + c52γ (by tc41 = c51tα + c52tγ) and c52 = c21α + c22γ + c11β + c12δ (by tc52 = c11 + c11tβ + c12tδ and c11 = c21tα + c22tγ) we have that c41 = 0 (taking into account the above equalities of the form cij = 0). Thus det(C) = 0, a contradiction. Lemma 4. The matrix M(t, 0, 0, 1, 1, 1) is irreducible. Proof. Assume that M(t, 0, 0, 0, 1, 1) is reducible. Then for some matrix C ∈ GL5(R) we have M(t, 0, 0, 1, 1, 1)C = C ( A D 0 B ) . (16) where A is an s × s matrix and B is a (5 − s) × (5 − s) matrix (0 < s < 5). As in the proof of Lemma 3, we can assume that s ≤ 5 − s, i. e. s = 1 or s = 2. Since the case s = 1 is trivial, we consider only the case s = 2. Let C = (cij)1≤i,j≤5, cp = (cp1, cp2) (p = 1, . . . , 5). Then from the equality (16) we obtain tc5 = c1A, c1 = c2A, c2 = c3A, tc3 = c4A, tc4 = c5A. (17) By Lemmas 1 and 2 rank(A) = 1. Since the matrix A is nilpotent we can assume, without loss of generality, that A = ( tα 1 + tβ tγ tδ ) , where α, β, γ, δ ∈ R. Substituting A in (17), we get the following equalities:              (tc51, tc52) = (c11tα + c12tγ, c11 + c11tβ + c12tδ), (c11, c12) = (c21tα + c22tγ, c21 + c21tβ + c22tδ), (c21, c22) = (c31tα + c32tγ, c31 + c31tβ + c32tδ), (tc31, tc32) = (c41tα + c42tγ, c41 + c41tβ + c42tδ), (tc41, tc42) = (c51tα + c52tγ, c51 + c51tβ + c52tδ). (18) From the equalities (18) we have c11 = c21 = 0 = c41 = c51, and c31 = c42γ (since tc31 = c41tα + c42tγ and c41 = 0), c22 = c31 (since c22 = c31 + c31tβ + c32tδ), c12 = 0 (since c12 = c21 + c21tβ + c22tδ and c21 = 0), c52γ = 0 (since tc41 = c51tα + c52tγ and c41 = c51 = 0), c52 = c31γ (since tc52 = c11 + c11tβ + c12tδ, c11 = c21tα + c22tγ and c11 = c21 = c12 = 0, c22 = c31). 184 Reducibility and irreducibility of monomial matrices If γ = 0 then c31 = 0; if γ 6= 0 then γ ∈ R∗ and hence c52 = 0, c31 = 0. Therefore, in both the cases det(C) = 0, a contradiction. Lemma 5. The matrix M(t, 0, 0, 1, 0, 1) is irreducible. Proof. Assume that M(t, 0, 0, 1, 0, 1) is reducible. Then for some matrix C ∈ GL5(R) we have M(t, 0, 0, 1, 0, 1)C = C ( A D 0 B ) . (19) where A is an s × s matrix and B is a (5 − s) × (5 − s) matrix (0 < s < 5). As in the proof of Lemma 3, we can assume that s ≤ 5 − s, i. e. s = 1 or s = 2. Since the case s = 1 is trivial, we consider only the case s = 2. Let C = (cij)1≤i,j≤5, cp = (cp1, cp2) (p = 1, . . . , 5). Then from the equality (19) we obtain tc5 = c1A, c1 = c2A, c2 = c3A, tc3 = c4A, c4 = c5A. (20) By Lemmas 1 and 2 rank(A) = 1. Since the matrix A is nilpotent we can assume, without loss of generality, that A = ( tα 1 + tβ tγ tδ ) , where α, β, γ, δ ∈ R. Substituting A in equality (20), we get              (tc51, tc52) = (c11tα + c12tγ, c11 + c11tβ + c12tδ), (c11, c12) = (c21tα + c22tγ, c21 + c21tβ + c22tδ), (c21, c22) = (c31tα + c32tγ, c31 + c31tβ + c32tδ), (tc31, tc32) = (c41tα + c42tγ, c41 + c41tβ + c42tδ), (c41, c42) = (c51tα + c52tγ, c51 + c51tβ + c52tδ). (21) From equality (21) we have c11 = c21 = c41 = 0, and c12 = 0 (since c12 = c21 + c21tβ + c22tδ and c21 = 0), c51 = 0 (since tc51 = c11tα + c12tγ and c11 = c12 = 0), c42 = 0 (since c42 = c51 + c51tβ + c52tδ and c51 = 0), c31 = c41α + c42γ = 0 (since tc31 = c41tα + c42tγ and c41 = c42 = 0). Therefore det(C) = 0, a contradiction. Lemma 6. The matrix M(t, 0, 1, 1, 0, 1) is irreducible. Bondarenko, Bortos, Dinis, Tylyshchak 185 Proof. Assume that M(t, 0, 1, 1, 0, 1) is reducible. Then for some matrix C ∈ GL5(R) we have M(t, 0, 1, 1, 0, 1)C = C ( A D 0 B ) . (22) where A is an s × s matrix and B is a (5 − s) × (5 − s) matrix (0 < s < 5). As in the proof of Lemma 3, we can assume that s ≤ 5 − s, i. e. s = 1 or s = 2. Since the case s = 1 is trivial, we consider only the case s = 2. Let C = (cij)1≤i,j≤5, cp = (cp1, cp2) (p = 1, . . . , 5). Then from the equality (22) we obtain tc5 = c1A, c1 = c2A, tc2 = c3A, tc3 = c4A, c4 = c5A. (23) By Lemmas 1 and 2 rank(A) = 1. Since the matrix A is nilpotent, we can assume, without loss of generality, that A = ( tα 1 + tβ tγ tδ ) , where α, β, γ, δ ∈ R. Substituting A in equality (23), we get              (tc51, tc52) = (c11tα + c12tγ, c11 + c11tβ + c12tδ), (c11, c12) = (c21tα + c22tγ, c21 + c21tβ + c22tδ), (tc21, tc22) = (c31tα + c32tγ, c31 + c31tβ + c32tδ), (tc31, tc32) = (c41tα + c42tγ, c41 + c41tβ + c42tδ), (c41, c42) = (c51tα + c52tγ, c51 + c51tβ + c52tδ). (24) From equality (24) we obtain that c11 = c41 = 0 = c31 and c12 = c21 (since c12 = c21 + c21tβ + c22tδ), c51 = c12γ (since tc51 = c11tα + c12tγ and c11 = 0), c21 = c32γ (since tc21 = c31tα + c32tγ and c31 = 0) c42 = c51 (since c42 = c51 + c51tβ + c52tδ), c42γ = 0 (since tc31 = c41tα + c42tγ and c31 = c41 = 0). If γ = 0 then c51 = 0 and c21 = 0. If γ 6= 0 then γ ∈ R∗, c42 = 0 and hence c51 = 0, c12 = 0 and c21 = 0. Therefore, det(C) = 0, a contradiction. 186 Reducibility and irreducibility of monomial matrices 3. Main result Theorem 2. Let R be a commutative local ring with radical Rad(R) = tR, t 6= 0, and let s1, . . . , sn ∈ {0, 1}. If 0 < n ≤ 6, then the matrix M(t, s1, . . . , sn) over R is irreducible if and only if n and s = ∑n i=1 si are coprime. Proof. The necessity part follows from Theorem 1. Let now n and ∑n i=1 si are coprime. Then the matrix M(t, s1, . . . , sn) is, up to cyclic permutations of si, one of the following: M(t, 0, 1), M(t, 0, 0, 1), M(t, 0, 0, 0, 1), M(t, 0, 0, 0, 0, 1), M(t, 0, 0, 0, 0, 0, 1), (25) M(t, 0, 1, 1), M(t, 0, 1, 1, 1), M(t, 0, 1, 1, 1, 1), M(t, 0, 1, 1, 1, 1, 1), (26) M(t, 0, 0, 0, 1, 1), M(t, 0, 0, 1, 1, 1), M(t, 0, 0, 1, 0, 1), M(t, 0, 1, 1, 0, 1). (27) The irreducibility of the matrices (25) are obvios. The matrices (26) are irreducible by [6], and the matrices (27) are irreducible by Lem- mas 3–6. The last theorem does not hold if n > 6. For example, if R is a local ring of length 2 and Rad(R) = tR (t 6= 0, t2 = 0), the matrix M = M(t, 0, 0, 0, 0, 1, 1, 1) (with n = 7 and s1 + · · · + s7 = 3 to be coprime) is reducible over R because, for C =             0 0 1 0 0 0 0 t 0 0 1 0 0 0 0 t 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 t 0 0 0 0 1             , C−1MC =             0 t 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 t 0 0 1 0 −t 0 0 0 0 0 1 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 t 0             . Bondarenko, Bortos, Dinis, Tylyshchak 187 References [1] V. M. Bondarenko, The similarity of matrices over rings of residue classes, Mathe- matics collection, Izdat. "Naukova Dumka”, Kiev, 1976, pp. 275–277 (in Russian). [2] V. N. Shevchenko, S. V. Sidorov, On the similarity of second-order matrices over the ring of integers, Izv. Vyssh. Uchebn. Zaved. Mat. (2006), no. 4, pp. 57–64 (in Russian). [3] A. Pizarro, Similarity classes of 3×3 matrices over a discrete valuation ring, Linear Algebra Appl. 54 (1983), pp. 29–51. [4] N. Avni, U. Onn, A. Prasad, L. Vaserstein, Similarity classes of 3 × 3 matrices over a local principal ideal ring, Comm. Algebra 37 (2009), N.8, pp. 2601–2615. [5] A. Prasad, P. Singla, and S. Spallone. Similarity of matrices over local rings of length two. (2012). http://arxiv.org/pdf/1212.6157.pdf. [6] P. M. Gudivok, O. A. Tylyshchak, On irreducible modular representations of finite p-groups over commutative local rings, Nauk. Visn. Uzhgorod. Univ. Ser. Math. (1998), N3, pp. 78–83 (in Ukrainian). Contact information V. M. Bondarenko Institute of Mathematics, Tereshchenkivska 3, 01601 Kyiv, Ukraine E-Mail: vit-bond@imath.kiev.ua URL: http://www.imath.kiev.ua M. Yu. Bortos Faculty of Mathematics, Uzhgorod National Univ., Universytetsyka str., 14, 88000 Uzhgorod, Ukraine E-Mail: bortosmaria@gmail.com R. F. Dinis Faculty of Mechanics and Mathematics, Kyiv Na- tional Taras Shevchenko Univ., Volodymyrska str., 64, 01033 Kyiv, Ukraine E-Mail: ruslanadinis@ukr.net A. A. Tylyshchak Faculty of Mathematics, Uzhgorod National Univ., Universytetsyka str., 14, 88000 Uzhgorod, Ukraine E-Mail: alxtlk@gmail.com Received by the editors: 20.10.2013 and in final form 20.10.2013.

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