Diagonalizability theorems for matrices over rings with finite stable range
We construct the theory of diagonalizability for matrices over Bezout ring with finite stable range. It is shown that every commutative Bezout ring with compact minimal prime spectrum is Hermite. It is also shown that a principal ideal domain with stable range 1 is Euclidean domain, and every sem...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2005 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Інститут прикладної математики і механіки НАН України
2005
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Diagonalizability theorems for matrices over rings with finite stable range / B. Zabavsky // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 151–165. — Бібліогр.: 35 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859629619646300160 |
|---|---|
| author | Zabavsky, B. |
| author_facet | Zabavsky, B. |
| citation_txt | Diagonalizability theorems for matrices over rings with finite stable range / B. Zabavsky // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 151–165. — Бібліогр.: 35 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | We construct the theory of diagonalizability for
matrices over Bezout ring with finite stable range. It is shown that
every commutative Bezout ring with compact minimal prime spectrum is Hermite. It is also shown that a principal ideal domain
with stable range 1 is Euclidean domain, and every semilocal principal ideal domain is Euclidean domain. It is proved that every
matrix over an elementary divisor ring can be reduced to "almost"
diagonal matrix by elementary transformations.
|
| first_indexed | 2025-12-07T13:09:08Z |
| format | Article |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2005). pp. 151 – 165
c© Journal “Algebra and Discrete Mathematics”
Diagonalizability theorems for matrices over
rings with finite stable range
Bogdan Zabavsky
Communicated by M. Ya. Komarnytskyj
Dedicated to Yu.A. Drozd on the occasion of his 60th birthday
Abstract. We construct the theory of diagonalizability for
matrices over Bezout ring with finite stable range. It is shown that
every commutative Bezout ring with compact minimal prime spec-
trum is Hermite. It is also shown that a principal ideal domain
with stable range 1 is Euclidean domain, and every semilocal prin-
cipal ideal domain is Euclidean domain. It is proved that every
matrix over an elementary divisor ring can be reduced to "almost"
diagonal matrix by elementary transformations.
Introduction
The aim of this note is to study the problem of diagonalizability for ma-
trices over rings with finite stable range. The theorem that each matrix
over a ring R is equivalent to some diagonal matrix was proved in the
case when R is the ring of integers in 1861 by Henry J.Stephen Smith [1].
It was consequently extended by Dickson [2], Wedderburn [3], van der
Waerden [4] and Jacobson [5] for various commutative and noncommuta-
tive Euclidean domains and commutative PIDs (principal ideal domains)
and later for noncommutative PIDs by O.Teichmuller [6] (then in some-
what sharper form in Asano [7] and Jacobson [8]). The theorem is also
known for an arbitrary PIR (principal ideal rings)[9].
Key words and phrases: finite stable range, elementary divisor ring, Hermite
ring, ring with elementary reduction of matrices, Bezout ring, minimal prime spec-
trum.
152 Diagonalizability theorems for matrices over rings...
Following Kaplansky[10] a ring R is said to be an elementary divisor
ring if every matrix over R is equivalent to a diagonal matrix. Kaplansky
proved that if R is an elementary divisor ring then every finitely presented
R-module is a direct sum of cyclic modules. In [11] the converse to
Kaplansky’s theorem for commutative ring is proved.
By [10] a ring is said to be Hermite if every 1 by 2 and 2 by 1 matrix
over it is equivalent to a diagonal matrix. Obviously, an elementary divi-
sor ring is Hermite and it is easy to see that an Hermite ring is Bezout (a
ring is a Bezout ring if every finitely generated 1-sided ideal is principal).
Examples that neither implication is reversible are provided by Gillman
and Henriksen in [12]. In [10] Kaplansky proved that a Bezout ring is
Hermite when all zero divisors of the ring are in the Jacobson radical es-
tablishing in particular the fact that all Bezout domains are Hermite [13].
Henriksen [14] changed Kaplansky’s hypothesis to the assumption that
the Jacobson radical contains a prime ideal and proved that the theorem
is still valid.
Adequate domains were introduced by Helmer in [15]. It had been
known that principal ideal domains were elementary divisor rings and
Helmer showed that the less restrictive hypothesis that an integral do-
main be adequate is sufficient. Kaplansky [10] began the consideration
of adequate rings with zero divisors by showing that the adequate ring
whose zero divisors are in the Jacobson radical is an elementary divisor
ring. M.Henriksen [16] proves that if R is a unit regular ring then every
matrix over R admits diagonal reduction. The diagonalizability question
for rectangular matrices was answered by Menal and Moncasi [17], who
showed that all rectangular matrices over given regular ring R admits a
diagonal reduction if and only if R is Hermite. If R is a separative regular
ring, then every square matrix over R admit a diagonal reduction [18].
In [19,20] it is proved that any left distributive ring R is an elementary
divisor ring if and only if R is invariant. Menal and Moncasi [17] showed
that any right Hermite and left Bezout ring is left Hermite. Further, the
stable range (in the sense of K-theory) of a right or left Hermite ring is
at most 2 [17].
In the present paper we construct the theory of diagonalizability for
matrices over Bezout ring with finite stable range.
In [14] Henriksen asked whether every commutative semilocal Bezout
ring is Hermite. This question was answered affirmatively in [11]. In
the situation of noncommutative ring this question was answered affir-
matively in [21].
Henriksen has raised the following question: If R is a commutative
Bezout ring with compact minimal prime spectrum, is R Hermite. De-
spite of results of [22], we show that every commutative Bezout ring with
B. Zabavsky 153
compact minimal prime spectrum is Hermite.
Following [23] a ring is said to be a right n-Hermite if every 1 by n
matrix over it is equivalent to a diagonal matrix. We show that a right
n-Hermite ring has stable range ≤ n. And if R is a right Bezout ring
with finite stable range n then R is a right n + 1-Hermite ring. We show
if R is left n-Hermite and right Bezout then R is right n-Hermite.
In [24] the problem of investigation of rings with elementary reduc-
tion of matrices is posed. A ring is said to be a ring with elementary
reduction of matrices if every matrix over R can be reduced to diagonal
form by using only elementary transformations. Clearly, every ring with
elementary reduction of matrices is an elementary divisor ring. But there
exists an elementary divisor ring which is not a ring with elementary re-
duction of matrices (e.g., the ring R[x, y]/(x2 + y2 + 1) where R is the
ring of reals)[24,25,35]. Obviously, every Euclidean domain is a ring with
elementary reduction of matrices. We show that over an elementary di-
visor ring, every matrix can be reduced to "almost" diagonal matrix by
elementary transformations.
1. Definitions
Furtheron R will always denote a ring (associative, but not necessary
commutative) with 1 6= 0. We shall write Rn for the ring of n by n
matrices with elements in R. By a unit of a ring we mean an element with
two-sided inverse. Units of Rn will be said to be unimodular. If b = ca
we say that a is a right divisor of b; equivalent conditions are b ∈ Ra and
Rb ⊆ Ra. We say that a is a total divisor of b if RbR ⊆ Ra ∩ aR, or
in words: everything in the two-sided ideal generated by b is right and
left divisible by a. It is observed that an element is not necessary a total
divisor of itself. If R is commutative then right, left and total divisibility
all coincide.
An n by m matrix A = (aij) is said to be diagonal if aij = 0 for all
i 6= j. We say that a matrix A admits diagonal reduction if there exist the
unimodular matrices P, Q such that PAQ is a diagonal matrix. We shall
call two matrices A and B over a ring R equivalent(notation A ∼ B) if
there exist the unimodular matrices P, Q such that B = PAQ. If every
matrix over R is equivalent to a diagonal matrix (dij) with the property
that every dii is a total divisor of di+1i+1 then R is an elementary divisor
ring. We recall that R is said to be right(left) Hermite if every 1 by 2
(2 by 1) matrix admits diagonal reduction, and if both, R is an Hermite
ring. If every 1 by n (n by 1) matrix admits diagonal reduction then R
is a right (left) n-Hermite ring.
A row (a1, . . . , an) over a ring R is called right unimodular if a1R +
154 Diagonalizability theorems for matrices over rings...
· · · + anR = R. If (a1, . . . an) is a right unimodular n-row over a ring R
then we say that (a1, . . . , an) is reducible if there exists an (n − 1)-row
(b1, . . . , bn−1) such that the (n − 1)-row (a1 + anb1, . . . an−1 + anbn−1) is
right unimodular. A ring R is said to have stable range n if n is the least
positive integer such that every right unimodular (n+1)-row is reducible.
This number is denoted by s.r.(R).
By a right (left) Bezout ring we mean a ring in which all finitely
generated right (left) ideals are principal, and by a Bezout ring a ring
which is both right and left Bezout.
A commutative ring R is said to be adequate if R is Hermite and for
a, b ∈ R with a 6= 0 there exist r, s ∈ R such that a = rs, rR + bR = R
and if a nonunit s′ divides s then s′R + bR 6= R. Under a right k-stage
division chain for elements a, b ∈ R, b 6= 0 we understand a sequence of
equalities:
a = bq1 + r1, b = r1q2 + r2, . . . , rk−2 = rk−1qk + rk.
A finite right division chain is by the definition a right k-stage division
chain for some k ∈ N. A norm over domain R is a function N : R −→
Z such that N(0) = 0 and N(a) > 0 for each a 6= 0. A domain R
is called a right n-Euclidean domain with respect to a norm N if for
any elements a, b ∈ R, b 6= 0 there exists a right k-stage division chain
such N(rk) < N(b). Obviously, any right 1-Euclidean domain is a right
Euclidean domain. A domain R is called a right ω-Euclidean domain with
respect to a norm N if for any elements a, b ∈ R, b 6= 0 there are k ∈ N
and a right k-stage division chain such that N(rk) < N(b). Any right
ω-Euclidean domain is a right Bezout domain. R is said to be regular if
for every a ∈ R there exists an x ∈ R such that axa = a. A regular ring
is said to be unit regular if for any a ∈ R there exists a unit u ∈ R such
that aua = a.
We denote by GLn(R) the group of units of Rn. We write GEn(R)
for the subgroup of GLn(R) generated by elementary matrices. The
Jacobson radical of a ring R will be denoted by J(R). Denote by U(R)
the group of units of R.
2. The space of minimal prime ideals
Let R be a commutative ring. By a minimal prime ideal of R we shall
mean a proper prime ideal that contains no smaller prime ideal. Thus,
for example, if R is an integer domain than (0) is the only minimal prime
ideal of R. Let min(R) be the minimal prime spectrum of R. If x ∈ R,
define D(x) = {P ∈ minR | x /∈ P}. Then the sets of the form D(x)
B. Zabavsky 155
form a basis for the Zariski topology of min R. When we say that R is
compact we mean that it is compact in this topology. Concerning rings
with compact minimal spectrum see [26-30].
In order to obtain a characterization of commutative Bezout ring with
compact minimal spectrum we need some preliminary results.
Theorem 1. [21,Theorem 1]. A commutative Bezout ring R is an Her-
mite ring if and only if R has stable range ≤ 2.
We denote by N the nilradical of R.
Proposition 1. If R is a commutative Bezout ring, then
s.r.(R) = s.r.(R/N).
Proof. By [31] s.r.(R/N) ≤ s.r.(R). Set R = R/N and let s.r.(R) ≤ n.
Let a1R + · · · + anR + an+1R = R then a1R + · · · + anR + an+1R = R.
Since s.r.(R) ≤ n, we obtain (a1 +an+1x1)R+ · · ·+(an +an+1xn)R = R.
Say (a1 + an+1x1)v1 + · · · + (an + an+1xn)vn = 1 for some v1, . . . , vn ∈
R. We get (a1 + an+1x1)v1 + · · · + (an + an+1xn)vn = 1 + m for some
x1, . . . xn, v1, . . . vn ∈ R and m ∈ N. Obviously, 1 + m ∈ U(R). Then
s.r.(R) ≤ n.
Theorem 2. Let R be a commutative Bezout ring with compact minimal
prime spectrum then R is Hermite.
Proof. We begin by showing that R/N is semihereditary. Every local-
ization of R/N is a semiprime valuation ring that is a valuation domain,
and it follows that every ideal of R/N is flat. Thus it will suffice by [32]
to prove that the classical quotient ring R/N is regular. But this follows
from [30,proposition 9] because every finitely generated faithful ideal of
R/N contains a non-zero divisor. Since every semihereditary Bezout ring
is Hermite [11, Theorem 2.4] then s.r.(R/N) ≤ 2. By proposition 1 then
s.r.(R) ≤ 2, i.e. R is a commutative Bezout ring with stable range ≤ 2.
By theorem 1 R is a Hermite ring.
3. A right n-Hermite ring
Proposition 2. If R is a right(left) n-Hermite ring then R has stable
range n.
Proof. Let a1R + · · · + anR + an+1R = R. Since R is a right n-Hermite
ring, (a1, . . . , an)P = (d, 0, . . . , 0) for some d ∈ R, P = (pij) ∈ GLn(R).
Let P−1 = (αij) ∈ GLn(R). We claim that (a1 + an+1αn1) + · · · + (an +
156 Diagonalizability theorems for matrices over rings...
an+1αnn) is a right unimodular row. We have (a1 + an+1αn1)p1n + · · ·+
(an +an+1αnn)pnn = a1p1n + · · ·+anpnn +an+1(αn1p1n + · · ·+αnnpnn) =
0 + an+1 · 1 = an+1 and (a1 + an+1αn1)p11 + · · · + (an + an+1αnn)pn1 =
a1p11 + · · ·+ anpn1 + an+1(αn1p11 + · · ·+αnnpn1) = d+ an+1 · 0 = d. and
an+1, d ∈ (a1+an+1αn1)R+ · · ·+(an+an+1αnn)R. Since (a1, . . . , an)P =
(d, 0, . . . , 0), we obtain a1R + · · · + anR = dR. On the other hand, we
have a1R+ · · ·+anR+an+1R = R then dR+an+1R = R. Since an+1, d ∈
(a1 +an+1αn1)R+ · · ·+(an +an+1αnn)R, we see that (a1 +an+1αn1)R+
· · ·+ (an + an+1αnn)R = R and s.r.(R) = n. Because the stable range of
a ring coincides with the stable range of its opposite ring [31, Theorem
2] the result also follows if R is left n-Hermite.
Proposition 3. If R is left n-Hermite and right Bezout ring then R is
right n-Hermite.
Proof. Let Ra1 + · · · + Ran = R, then P
a1
...
...
an
=
1
0
...
0
for some
P ∈ GLn(R). Clearly,
a1
...
...
an
= P−1
1
0
...
0
. Let P−1 = (pij), then
a1 = p11, . . . , an = pn1. Now we obtain that every left unimodular n-
column is the first column of unimodular n by n matrix over R. We first
prove the analogous result for rows. If a1R + · · ·+ anR = R then a1u1 +
· · · + anun = 1 for some u1, . . . un ∈ R and there exists an unimodular
n by n matrix Q of the form Q =
u1
...
un
∗
. Clearly, (a1, . . . , an)Q =
(1, b2, . . . , bn). If
P =
1 −b2 −b3 · · · −bn
0 1 0 · · · 0
0 0 1 · · · 0
...
...
...
. . .
...
0 0 0 · · · 1
,
then P ∈ GEn(R) and (a1, . . . , an)QP = (1, 0, . . . , 0). The row (a1, . . . , an)
is the first row of the matrix P−1Q−1 ∈ GLn(R). Now we prove that R
is right n-Hermite. Let a1, . . . , an ∈ R then, since R is right Bezout
B. Zabavsky 157
a1R+ · · ·+anR = dR, say ai = da0
i , (i = 1, . . . , n), d = a1u1 + · · ·+anun.
We get d(a0
1 + · · · + a0
nun − 1) = 0, so for some c ∈ R such that dc = 0
we have a0
1R + · · ·+ a0
nR + cR = R. It follows from proposition 2 that R
has stable range ≤ n thus (a0
1 + cv1)R + · · · + (a0
n + cvn)R = R for some
v1, . . . vn ∈ R. By the above we can find a unimodular matrix Q of the
form
Q =
(
a0
1 + cv1 · · · a0
n + cvn
∗
)
.
Clearly, (a1, . . . , an) = (d, 0, . . . , 0)Q then (a1, . . . , an)Q−1 = (d, 0, . . . , 0)
so R is right n-Hermite.
Proposition 4. If R is a right n-Hermite ring then R is right Bezout.
Proof. Let a, b ∈ R, then since R is a right n-Hermite, (a, b, 0, . . . , 0)P =
(d, 0, . . . , 0) for some d ∈ R where P = (pij) ∈ GLn(R). We get ap11 +
bp21 = d.
Let P−1 = (αij) ∈ GLn(R). Clearly, (a, b, . . . , 0) = (d, 0, . . . , 0)P−1,
then a = dα11, b = dα21 and aR ⊂ dR, bR ⊂ dR. Then aR + bR ⊂ dR
and aR + bR = dR. Therefore, R is right Bezout.
Corollary 1. Let R be a right Bezout ring with finite stable range n, then
for any a1, . . . , am ∈ R where m ≥ n+1 there exists a unimodular matrix
P ∈ GEm(R) such that (a1, . . . , am)P = (d, 0, . . . , 0) for some d ∈ R.
Proof. We first prove that any right unimodular row of length m over
R can be completed to a unimodular matrix. If a1R + · · · + amR = R,
then there exists m − 1-row (v1, . . . , vm−1) such that (a1 + amv1)R +
· · · + (am−1 + amvm−1)R = R. There exist u1, . . . , um−1 ∈ R such that
(a1 + amv1)u1 + · · · + (am−1 + amvm−1)um−1 = 1.
Set
P1 =
1 0 . . . 0 0
0 1 . . . 0 0
...
...
. . .
...
...
v1 v2 . . . vm−1 1
∈ GEn(R);
P2 =
1 0 . . . 0 0
0 1 . . . 0 u1(1 − am)
0 1 . . . 0 u2(1 − am)
...
...
. . .
...
...
0 0 . . . 1 um−1(1 − am)
0 0 . . . 0 1
∈ GEn+1(R).
We see that for a row (a1, . . . , am)P1P2 there exists a matrix P3 ∈
GEm(R) such that (a1, . . . , am)P1P2P3 = (1, 0, . . . , 0). Thus we obtain
158 Diagonalizability theorems for matrices over rings...
a matrix P ∈ GEm(R) such that (a1, . . . , am)P = (1, 0, . . . , 0). Then
(a1, . . . , am) is the first row of the matrix P−1. Since R is a right Bezout
ring, for any a1, . . . , am ∈ R there exists d ∈ R such that a1R + · · · +
amR = dR. Say a1u1 + · · · + amum = d, a1 = da0
1, . . . , am = da0
m. From
these relations we get d(a0
1u1 + · · ·+ a0
mum − 1) = 0, so that a0
1R + · · ·+
a0
mR + cR = R for some c ∈ R such that dc = 0. Since s.r.(R) < m, we
see that (a0
1 + cv1)R + · · ·+ (a0
m + cvm)R = R, where v1, . . . , vm ∈ R. By
the above we can find a unimodular matrix P ∈ GEm(R) of the form
P =
(
a0
1 + cv1 · · · a0
m + cvm
∗
)
.
Clearly, (a1, . . . , am)P−1 = (d, 0, . . . , 0).
4. Bass’ first stable range condition
Bass’ lowest (the first) stable range condition [34] asserts the following:
if a and b in R satisfy Ra + Rb = R then there exists t in R with a + tb
left invertible. More exactly this is the "left" version of the condition,
and there is a symmetric "right" version, but the two versions are in the
fact equivalent [34].
We shall now discuss the question of the uniqueness of the generators
of principal right ideals. If a = bu, where u is a unit, we say that a and
b are right associates. Clearly, associate elements are right multiples of
each other or they generate the same principal right ideals. We raise the
converse question: If aR = bR, are a and b necessarily right associate?
It is well known that the answer is affirmative if there are no divisors of
0 [8]. Kaplansky [10] extended this result to the ring in which all right
divisors of 0 are in the radical. We shall prove this for ring with stable
range 1.
Proposition 5. Let R be a ring with stable range 1. Then aR = bR
implies that a, b are right associate.
Proof. We have a = by, b = ax so a = axy. If a = b = 0 there is nothing
to prove. Otherwise a(1 − xy) = 0. Let 1 − xy = c, then xR + cR =
R, ac = 0. Since s.r.(R) = 1, we have x+ cv = u ∈ U(R) for v ∈ R. Thus
ax + acv = au. Then ax − au = b and bu−1 = a.
Corollary 2. Let R be a ring of stable range 1. If A1, A2 ∈ Rn are
matrices which are right multiples of each other. Then A1, A2 are right
associate.
Proof. Since for any natural number n s.r.(R) = 1 if and only if so is Rn
[34, Theorem 2.4], by theorem 5 the proof of corollary is obvious.
B. Zabavsky 159
Proposition 6. Let R be a right Bezout ring of stable range 1. Then
for any a, b ∈ R there exist x ∈ R, d ∈ R such that a + bx = d and
aR + bR = dR.
Proof. Since R is a right Hermite ring [21, Theorem 2], for any a, b there
exist δ, a0, b0 ∈ R such that a = δa0, b = δb0 and a0R + b0R = R. Since
s.r.(R) = 1, there exist x ∈ R, u ∈ U(R) such that a0 + b0x = u. Then
a + bx = δu. Obviously, δuR = δR. Set d = δu, then a + bx = d and
aR + bR = dR.
Proposition 7. Let R be a right Bezout domain with stable range 1.
Then R is a right 2-Euclidean domain.
Proof. Let N be a function R −→ Z such that N(0) = 0 and N(a) = 1 for
each a 6= 0. Let a, b ∈ R, b 6= 0, by proposition 6, there exist x ∈ R, d ∈ R
such that a + bx = d and aR + bR = dR, then a = da0, b = db0 for some
a0, b0 ∈ R. Thus a = b · (−x) + d, b = db0 + 0 and N(0) < N(b).
Proposition 8. Let R be a principal ideal domain with stable range 1.
Then R is Euclidean domain.
Proof. Let |a| denote the number of prime factors of a ∈ R \ 0 in the
factorization of a into prime factors. Obviously, |a| ≥ 0 and |ab| = |a|+|b|.
By proposition 6 there exist x ∈ R, d ∈ R for a, b ∈ R, b 6= 0 such that
a + bx = d and aR + bR = dR. Let b = db0 for b0 ∈ R. If |d| < |b| then
a = b(−x) + d. If |d| = |b| then |b0| = 0 and b0 ∈ U(R). Since a = da0 for
some a0 ∈ R, we see that a = bb−1
0
a0, i.e. aR ⊂ bR.
Since every semilocal ring is a ring with stable range 1, we have
Corollary 3. Every semilocal principal ideal domain is a Euclidean do-
main.
5. A ring with elementary reduction of matrices
Recall that a ring is said to be a ring with elementary reduction of ma-
trices if every matrix can be reduced to a diagonal form by using only
elementary transformations [24].
Proposition 9. Let R be a right Bezout ring and for any elements a, b ∈
R there exists a unimodular matrix Q ∈ GE2(R) such that Q
(
a
b
)
=
(
d
0
)
for d ∈ R. Then there exists a unimodular matrix P ∈ GE2(R)
such that (a, b)P = (c, 0) for any a, b ∈ R.
160 Diagonalizability theorems for matrices over rings...
Proof. Let Ra + Rb = R. Then there exists unimodular 2 by 2 matrix
Q ∈ GE2(R) such that Q
(
a
b
)
=
(
1
0
)
. Then we have
(
a
b
)
= Q−1
(
1
0
)
(1)
where Q−1 ∈ GE2(R). Let Q−1 =
(
q11 q12
q21 q22
)
. By (1) we have a =
q11, b = q21. Then
(
a
b
)
is the first column of the matrix Q−1 ∈ GE2(R)
i.e. any left unimodular column of length 2 over R can be completed to a
unimodular matrix in GE2(R). Since R is left Hermite and right Bezout,
R is a right Hermite ring [17, Proposition 8].
Let aR + bR = R, then there exist u, v ∈ R such that au + bv = 1.
We know that the left unimodular column
(
u
v
)
can be completed to
a unimodular matrix U ∈ GE2(R). Then (a, b)U = (1, c) for a suitable
element c ∈ R. We see that for the row (a, b)U there exists a unimodular
matrix V ∈ GE2(R) such that (a, b)UV = (1, 0). Thus we obtain a
unimodular matrix P ∈ GE2(R) such that (a, b)P = (1, 0). Then (a, b) is
the first row of the matrix P−1 ∈ GE2(R).
Since R is Hermite, for every pair of elements a, b ∈ R the following
holds: there exist d, a′, b′ ∈ R such that a = da′, b = db′ and a′R + b′R =
R. By the above argument we can find a unimodular matrix P ∈ GE2(R)
such that (a′, b′)P = (1, 0). Clearly, (a, b)P = (d, 0), which finishes the
proof.
Corollary 4. If R is a right ω-Euclidean Bezout ring then R is a left
ω-Euclidean ring.
Proof. If R is a right ω-Euclidean ring then for any elements a, b ∈ R there
exists a unimodular matrix Q ∈ GE2(R) such that Q
(
a
b
)
=
(
d
0
)
[35, Proposition 1]. By proposition 9 there exists a unimodular matrix
P ∈ GE2(R) such that (a, b)P = (c, 0) for any a, b ∈ R.
By [35, Proposition 4] R is a left ω-Euclidean ring.
Theorem 3. Let R be an elementary divisor ring then, for any n by m
matrix A (n > 2, m > 2) we can find unimodular matrices P ∈ GEn(R)
B. Zabavsky 161
and Q ∈ GEm(R) such that
PAQ =
ǫ1 0 · · · 0 0 0 0
0 ǫ2 · · · 0 0 0 0
...
... · · ·
...
...
...
...
0 0 · · · ǫs 0 0 0
0 0 · · · 0
0 0 · · · 0 A0
0 0 · · · 0
,
where ǫi is a total divisor of ǫi+1, 1 ≤ i ≤ s − 1 and A0 – 2 by k or k by
2 matrix.
Proof. Since R is an elementary divisor ring, we can find a unimodular
matrix P1 such that
P1A =
(
a′11 · · · a′2m
∗
)
where a′11R + · · · + a′1mR = ǫ1R, where ǫ1 is a total divisor of all the
elements of P1A. Let P1 = (p′ij) ∈ GEn(R). Obviously, p′11R + · · · +
p′1nR = R. Since any elementary divisor ring is Hermite, s.r.(R) = 2.
Since n > 2 by [33, Proposition 1], the right unimodular row (p′12, . . . , p
′
1n)
can be completed to a unimodular matrix H1 ∈ GEn(R). Then
H1A =
(
a′11 · · · a′1m
∗ · · · ∗
)
.
Since m > 2, there exists a unimodular matrix S1 ∈ GEm(R) such that
H1AS1 =
(
ǫ1 0 · · · 0
∗
)
.
We may now use elementary transformations to sweep out the first column
of H1AS1 and we obtain
ǫ1 0 · · · 0
0
... A1
0
,
where ǫ1 is still a total divisor of every elements of A1. Proceeding in this
way we complete the reduction.
162 Diagonalizability theorems for matrices over rings...
Theorem 4. Let R be an elementary divisor ring. Then for every n
by m matrix A, where m − n = 2 we can find a unimodular matrices
P ∈ GLn(R), Q ∈ GEm(R) such that PAQ is a diagonal matrix
diag(d1, . . . , dr, 0, . . . , 0),
where di is a divisor of di+1, i = 1, 2, . . . , r − 1.
Proof. By theorem 3 we need only to consider the case of a 2 by 4 matrix
A. Since R is an elementary divisor ring, there exists a unimodular matrix
P ∈ GL2(R) such that
PA =
(
a11 a12 a13 a14
a21 a22 a23 a24
)
,
with a11R + a12R + . . . + a14R = ǫR, where ǫ is a total divisor of all the
elements of PA. Since any elementary divisor ring is Hermite, R is a ring
with stable range ≤ 2. By corollary 1 there exist a unimodular matrix
Q ∈ GE4(R) such that
PAQ =
(
ǫ 0 0 0
∗ ∗ ∗ ∗
)
.
We may now use elementary transformations to sweep out the first column
of PAQ and we obtain
(
ǫ 0 0 0
0 b22 b23 b24
)
,
where ǫ is a total divisor of every elements b22, b23, b24.
By corollary 1, there exists a unimodular matrix W ∈ GE3(R) such
that (b22, b23, b24)W = (b, 0, 0).
Then
(
ǫ 0 0 0
0 b22 b23 b24
)
1 0 0 0
0
0 W
0
=
(
ǫ 0 0 0
0 b 0 0
)
,
where, obviously, ǫ is a divisor of b.
Proposition 10. Let R be a commutative adequate ring then for ev-
ery nonsingular n by n matrix A we can find unimodular matrices P ∈
GEn(R), Q ∈ GLn(R) such that
PAQ =
ǫ1 0 · · · 0
0 ǫ2 · · · 0
...
...
. . .
...
0 · · · 0 ǫn
,
B. Zabavsky 163
where ǫi is a divisor of ǫi+1, 1 ≤ i ≤ n − 1.
Proof. Let n = 2, without loss of generality we may change notations
and assume that the greatest common divisor off all elements of A is
1. Since R is Hermite, s.r.(R) = 2 and we find a unimodular matrix
Q1 ∈ GL2(R) such that AQ1 =
(
a 0
b c
)
. Since A is a nonsingular,
c 6= 0. Write c = rs where rR + aR = R and if a nonunit element s′
divides s then s′R + aR 6= R. Then, obviously, (ra + b)R + cR = R.
Multiplying the first row of a matrix AQ1 by r and adding it to the
second row, we obtain the matrix
A1 =
(
a 0
ra + b c
)
.
Since (ra+ b)R+ cR = R, there exist a unimodular matrix Q2 ∈ GL2(R)
such that
A1Q2 =
(
∗ ∗
1 0
)
.
The matrix A1Q2 is reducible by elementary transformations to the form
(
1 0
0 ∆
)
. Application of theorem 3 completes the proof of this propo-
sition.
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Contact information
B. Zabavsky Ivan Franko Lviv National University
E-Mail: b_zabava@franko.lviv.ua
Received by the editors: 11.06.2004
and in final form 21.03.2005.
|
| id | nasplib_isofts_kiev_ua-123456789-156607 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T13:09:08Z |
| publishDate | 2005 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Zabavsky, B. 2019-06-18T17:49:22Z 2019-06-18T17:49:22Z 2005 Diagonalizability theorems for matrices over rings with finite stable range / B. Zabavsky // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 151–165. — Бібліогр.: 35 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/156607 We construct the theory of diagonalizability for matrices over Bezout ring with finite stable range. It is shown that every commutative Bezout ring with compact minimal prime spectrum is Hermite. It is also shown that a principal ideal domain with stable range 1 is Euclidean domain, and every semilocal principal ideal domain is Euclidean domain. It is proved that every matrix over an elementary divisor ring can be reduced to "almost" diagonal matrix by elementary transformations. Dedicated to Yu.A. Drozd on the occasion of his 60th birthday en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Diagonalizability theorems for matrices over rings with finite stable range Article published earlier |
| spellingShingle | Diagonalizability theorems for matrices over rings with finite stable range Zabavsky, B. |
| title | Diagonalizability theorems for matrices over rings with finite stable range |
| title_full | Diagonalizability theorems for matrices over rings with finite stable range |
| title_fullStr | Diagonalizability theorems for matrices over rings with finite stable range |
| title_full_unstemmed | Diagonalizability theorems for matrices over rings with finite stable range |
| title_short | Diagonalizability theorems for matrices over rings with finite stable range |
| title_sort | diagonalizability theorems for matrices over rings with finite stable range |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/156607 |
| work_keys_str_mv | AT zabavskyb diagonalizabilitytheoremsformatricesoverringswithfinitestablerange |