Elementary reduction of matrices over rings of almost stable range 1
In this paper we consider elementary reduction of matrices over rings of almost stable range 1.
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irk-123456789-1885212023-03-04T01:27:12Z Elementary reduction of matrices over rings of almost stable range 1 Zabavsky, B. Romaniv, A. Kysil, T. In this paper we consider elementary reduction of matrices over rings of almost stable range 1. 2020 Article Elementary reduction of matrices over rings of almost stable range 1 / B. Zabavsky, A. Romaniv, T. Kysil // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 271–276. — Бібліогр.: 6 назв. — англ. 1726-3255 DOI:10.12958/adm1211 2010 MSC: 13F99. http://dspace.nbuv.gov.ua/handle/123456789/188521 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we consider elementary reduction of matrices over rings of almost stable range 1. |
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Zabavsky, B. Romaniv, A. Kysil, T. |
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Zabavsky, B. Romaniv, A. Kysil, T. Elementary reduction of matrices over rings of almost stable range 1 Algebra and Discrete Mathematics |
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Zabavsky, B. Romaniv, A. Kysil, T. |
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Zabavsky, B. |
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Elementary reduction of matrices over rings of almost stable range 1 |
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Elementary reduction of matrices over rings of almost stable range 1 |
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Elementary reduction of matrices over rings of almost stable range 1 |
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Elementary reduction of matrices over rings of almost stable range 1 |
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Elementary reduction of matrices over rings of almost stable range 1 |
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elementary reduction of matrices over rings of almost stable range 1 |
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Інститут прикладної математики і механіки НАН України |
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Elementary reduction of matrices over rings of almost stable range 1 / B. Zabavsky, A. Romaniv, T. Kysil // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 271–276. — Бібліогр.: 6 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT zabavskyb elementaryreductionofmatricesoverringsofalmoststablerange1 AT romaniva elementaryreductionofmatricesoverringsofalmoststablerange1 AT kysilt elementaryreductionofmatricesoverringsofalmoststablerange1 |
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2025-07-16T10:36:58Z |
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2025-07-16T10:36:58Z |
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1837799539685195776 |
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“adm-n2” — 2020/7/8 — 9:54 — page 271 — #131
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 2, pp. 271–276
DOI:10.12958/adm1211
Elementary reduction of matrices over rings
of almost stable range 1
B. Zabavsky, A. Romaniv, and T. Kysil
Communicated by V. V. Kirichenko
Abstract. In this paper we consider elementary reduction
of matrices over rings of almost stable range 1.
Introduction
One of important classes of rings that are widely used in modern algebra
is the class of Euclidean rings. The existence of the Euclid algorithm
is a great benefit for solving problems related to matrix calculations.
Naturally, this fact is a reason of interest to study rings, such that every
matrix over them can be reduced to the canonical diagonal form using only
elementary operations. These rings are called the rings with elementary
matrix reduction. However, there are elementary divisor rings that are
not rings with elementary matrix reduction [4]. In the same time, as it
will be shown in what follows, the use of elementary operations is crucial
in elementary matrix reduction.
A ring is a right (left) Bezout ring if every finite generated right (left)
ideal is principal.
Two matrices A and B over a ring R are said to be equivalent if there
exist invertible matrices P and Q such that B = PAQ. A matrix A admits
diagonal reduction if A is equivalent to a diagonal matrix D = (di) where
di is total divisor of di+1, i.e. Rdi+1R ⊂ diR ∩Rdi.
2010 MSC: 13F99.
Key words and phrases: elementary reduction of matrices, diagonal reduction
of matrices, rings of almost stable range 1.
https://doi.org/10.12958/adm1211
“adm-n2” — 2020/7/8 — 9:54 — page 272 — #132
272 Elementary reduction of matrices
A ring R is called an elementary divisor ring if every matrix over R
has a diagonal reduction [1].
We denote by Rn the ring of all n×n matrices over R and by GLn(R)
its group of unites. We write GEn for the subgroup of GLn(R) generated
by the elementary matrices. Denote by (a, b) the greatest common divisor
of elements a, b of commutative ring R.
The Jacobson radical of a ring R is denoted by J(R).
A ring R is called a ring of stable range 1 if for any elements a, b ∈
R the equality aR + bR = R implies that there is some x ∈ R such
that (a + bx)R = R. If for any elements a, b, c of a ring R the equality
aR + bR + cR = R implies that there are some elements x, y ∈ R such
that (a+ cx)R+ (b+ cy)R = R then we say that the stable range is equal
to 2.
The main results
We need the following results to start.
Theorem 1 ([4]). Let R be an elementary divisor ring. Then for any
n × m matrix A (n > 2,m > 2) one can find such invertible matrices
P ∈ GEn(R) and Q ∈ GEm(R) that
ε1 0 . . . 0 0 . . . 0
0 ε2 . . . 0 0 . . . 0
. . . . . .
.
.
. . . . . . . . . . . . .
0 0 . . . εs 0 . . . 0
0 0 . . . 0
. . . . . . . . . . . . A0
0 0 . . . 0
,
where εi is the total divisor of εi+1, 1 6 i 6 s− 1 and A0 is 2× k or k× 2
matrix for some k ∈ N .
Theorem 2 ([4]). Let R be an elementary divisor ring. Then for any
n×m matrix where m−n = 2 there exist invertible matrices P ∈ GEn(R)
and Q ∈ GEm(R) such that
ε1 0 . . . 0 0 . . . 0
0 ε2 . . . 0 0 . . . 0
. . . . . .
.
.
. . . . . . . . . . . . .
0 0 . . . εs 0 . . . 0
0 0 . . . 0 0 . . . 0
. . . . . . . . . . . . . . . . . . . . .
0 0 . . . 0 0 . . . 0
.
“adm-n2” — 2020/7/8 — 9:54 — page 273 — #133
B. Zabavsky, A. Romaniv, T. Kysil 273
Definition 1. An element a 6= 0 of a commutative ring R is called an
element of almost stable range 1 if the stable range of R/aR is equal to 1.
If all nonzero elements of a ring R are elements of almost stable range 1,
then we say that R is a ring of almost stable range 1.
The following results established in [5] play a key role in our work.
Proposition 1. Suppose R is a commutative Bezout ring. If R has stable
range 1, then R is a ring of almost stable range 1.
Theorem 3. Suppose R is a commutative ring of almost stable range 1.
Whenever a /∈ J(R) and aR + bR + cR = R there is r ∈ R such that
aR+ (b+ cr)R = R. In particular R is a ring of stable range 2.
Theorem 4. Suppose R is a commutative ring of almost stable range 1.
Then R is a Bezout ring if and only if R is an elementary divisor ring.
By [6] we have a next result.
Proposition 2. Let R be a commutative ring. The following properties
are equivalent:
1) R is a ring of almost stable range 1;
2) For any elements a, b ∈ R such that aR+ bR+ cR = R where a 6= 0
exist element r ∈ R such that
aR+ (b+ cr)R = R.
Definition 2. A ring R is said to be a ring of almost right(left) stable
range 1 if for any elements a, b, c ∈ R where a 6= 0 such that aR+bR+cR =
R (Ra+Rb+Rc = R) there exist an element r ∈ R such that
aR+ (b+ cr)R = R (Ra+R(b+ rc) = R).
Open question. Is a ring of almost right(left) stable range 1 a ring of
almost left(right) stable range 1?
Proposition 3. Let R be a right(left) Bezout ring of stable range 1 then
R is of almost right(left) stable range 1.
Proof. Let aR + bR + cR = R, a 6= 0 and bR + cR = R. Since R is
a ring of stable range 1 then b = db1, c = dc1 where b1R + c1R = R
[4]. Then (b1 + c1λ)R = R, i.e. b1 + c1λ = u – invertible element of R.
Since (b + cλ)R = dR and aR + bR + cR = R then aR + dR = R i.e.
aR+ (b+ cλ)R = R. The proposition is proved.
“adm-n2” — 2020/7/8 — 9:54 — page 274 — #134
274 Elementary reduction of matrices
By [2] we have the following.
Proposition 4. Let R be a commutative ring of almost stable range 1.
Then R2 is a ring of almost right and left stable range 1.
Theorem 5. Over a commutative Bezout domain of almost stable range 1
every k × (k + 2) and (k + 2)× k matrix, where k > 2, admits diagonal
reduction by elementary transformations.
Proof. We prove the theorem by induction on the number of rows. Let
A be a 2× 4 matrix. Without loss of generality we may change notation
and assume that the greatest common divisor of all elements is 1 [4]. By
Theorem 1, the matrix A can be reduced by elementary transformations
on the left to the form
A1 =
(
a 0 0 0
b c d k
)
.
If b = 0, by [4], the matrix A can be reduced by elementary transfor-
mations to a diagonal form.
If b 6= 0, write (a, b, c) = δ, a = a0δ, b = b0δ, c = c0δ and (a0, b0, c0) = 1.
Write c0 = rs with the property asserted above. Then
(ra+ b, c) = (a, b, c) = δ.
Multiplying the first row of a matrix A1 by r and adding it to the
second row, we obtain the matrix
A2 =
(
a 0 0 0
ra+ b c d k
)
.
Since (ra+ b, c) = (a, b, c) and (a, b, c, d, k) = 1, we have
(ra+ b, c, d, k) = 1.
By [4], the matrix A2 is reducible by elementary transformations to
the form
A3 =
(
1 0 0 0
∗ ∗ ∗ ∗
)
.
It is obvious that A3 is reducible by elementary transformations to a
diagonal form. Induction on the number of rows completes the proof.
“adm-n2” — 2020/7/8 — 9:54 — page 275 — #135
B. Zabavsky, A. Romaniv, T. Kysil 275
Theorem 6. Let R be commutative Bezout domain of almost stable
range 1, then for any nonsingular matrix of size n we can find such
unimodular matrices P ∈ GEn(R) and Q ∈ GLn(R), that
PAQ =
ε1 0 . . . 0
0 ε2 . . . 0
...
...
. . .
...
0 0 . . . εn
,
where εi is divisor εi+1, 1 6 i 6 n− 1.
Proof. Since commutative Bezout domain of almost stable range 1 is ele-
mentary divisor domain (see Theorem 4), then, it is obvious, by Theorem 1,
to prove the statement it is sufficient to consider the case of a matrix
of size 2. Without loss of generality, we can suppose, that the greatest
common divisor of elements of the matrix equal 1 [4]. Since R is Hermite
ring, then by corollary 2.1.1 [4] stable range of R is less or equal to 2.
Then for any matrix A we can find a matrix Q1 ∈ GL2(R) such that
AQ1 =
(
a 0
b c
)
.
Note, that
aR+ bR+ cR = R.
Since A ia nonsingular, then a 6= 0 and c 6= 0. Due to the fact that R is
commutative Bezout domain of almost stable range 1,
(ra+ b)R+ cR = R.
If we multiply the first row of AQ1 by r and add to the second row of
AQ1, we obtain the matrix
(
a 0
ra+ b c
)
.
Since
(ra+ b)R+ cR = R,
then exists a unimodular matrix Q2 ∈ GL2(R) such that
(
a 0
ra+ b c
)
Q2 =
(
∗ ∗
1 0
)
.
“adm-n2” — 2020/7/8 — 9:54 — page 276 — #136
276 Elementary reduction of matrices
Obviously, the matrix
(
a 0
ra+ b c
)
Q2
can be reduced via elementary operations to
(
1 0
0 ac
)
.
References
[1] Kaplansky I. Elementary divisors and modules. Trans. Amer. Math. Soc. 166,
1966, pp.464–491.
[2] Shchedryk V. Bezout Rings of Stable Range 1.5 Ukrainian Math. J., 67 (6), 2015,
pp.960–974.
[3] Zabavsky B. Reduction of matrices over right Bezout rings with finite stable rank.
Matematychni Studii, 16 (2), 2001, 115–116.
[4] Zabavsky B. Diagonal reduction of matrices over rings. Mathematical Studies,
Monograph series, vol. XVI, VNTL Publishers 2012, 249p.
[5] McGovern W. Bezout rings with almost stable range 1 are elementary divisor rings.
J. Pure and Appl. Algebra, 212, 2007, pp.340–348.
[6] Anderson D., Juett J. Stable range and almost stable range. J. Pure and Appl.
Algebra, 216, 2012, pp.2094–2097.
Contact information
Bohdan Zabavsky Ivan Franko National University of L’viv, 1
Universytetska str.,79000, L’viv, Ukraine
E-Mail(s): zabavskii@gmail.com
Andriy Romaniv Pidstryhach Institute for Applied Problems of
Mechanics and Mathematics NAS of Ukraine,
3b Naukova Str., L’viv, 79060, Ukraine
E-Mail(s): romaniv_a@ukr.net
Tatyana Kysil Khmelnytskyi National University, 11
Institutska str., 29000, Khmelnitsky, Ukraine
E-Mail(s): kysil_tanya@ukr.net
Received by the editors: 04.07.2018.
B. Zabavsky, A. Romaniv, T. Kysil
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