Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides
It is proved that each matrix over Bezout domain of stable range 1 with Dubrovin's condition, in which every maximal nonprincipal ideals are tho-sides ideals, is equivalent to diagonal one with right total division of diagonal elements.
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Інститут прикладної математики і механіки НАН України
2012
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| Cite this: | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides / T. Kysil, B. Zabavskiy, O. Domsha // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 230–235. — Бібліогр.: 10 назв. — англ. |
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| author | Kysil, T. Zabavskiy, B. Domsha, O. |
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| citation_txt | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides / T. Kysil, B. Zabavskiy, O. Domsha // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 230–235. — Бібліогр.: 10 назв. — англ. |
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| description | It is proved that each matrix over Bezout domain of stable range 1 with Dubrovin's condition, in which every maximal nonprincipal ideals are tho-sides ideals, is equivalent to diagonal one with right total division of diagonal elements.
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 14 (2012). Number 2. pp. 230 – 235
c© Journal “Algebra and Discrete Mathematics”
Reduction of matrices
over Bezout domains of stable range 1
with Dubrovin’s condition in which maximal
nonprincipal ideals are two-sides
Tetyana Kysil, Bogdan Zabavskiy, Olga Domsha
Communicated by V. V. Kirichenko
Abstract. It is proved that each matrix over Bezout domain
of stable range 1 with Dubrovin’s condition, in which every maximal
nonprincipal ideals are tho-sides ideals, is equivalent to diagonal
one with right total division of diagonal elements
Introduction
I.Kaplansky began systematic study of elementary divisors rings [1].
Many articles about commutative elementary divisors rings are already
written and many are still under research. But as to non-commutative
elementary divisors rings not much is known. It’s classical that principal
ideals domain is elementary divisors domain [2].
It’s necessary to mention the result of P.Cohn [3] who proved, that
the right principal Bezout domain is the domain, over which every matrix
is equivalent to the diagonal matrix with the condition of the right total
division of the diagonal elements. It’s also important to consider the
work of Dubrovin, who proved, that the semi-local semi-prime Bezout
ring is the elementary divisors ring if and only if for any element a ∈ R
there exists the element b ∈ R R such that RaR = bR = Rb (today
Key words and phrases: Bezout domain, domain of stable range 1, Dubrovin’s
condition, maximal nonprincipal ideal, right total division.
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T. Kysil, B. Zabavskiy, O. Domsha 231
this condition is called Dubrovin’s condition). In [5] it is shown, that the
distributive Bezout domain is the elementary divisors ring if and only
if it’s duo-domain (as a distributive Bezout domain we understand the
Bezout domain, where every maximal one-sided ideal is two-sided).
During last year K-theory was actively applied for the research of
elementary divisors rings. There particularly important is the use of such
invariant as the stable range of rings [8, 7]. Since the semi-local ring is
the ring of the stable range 1, then taking into consideration the above-
mentioned the next step in the research is Bezout domains of the stable
range 1 with Dubrovin’s condition. Generally, Bezout domains differ from
the principal ideal domains by existence of non-principal two-sided ideals.
Therefore study of the influence of the structure of non-principal right
ideals on possibility of the diagonal reduction is actual.
1. Preliminaries
For the future as a ring we shall understand the associative ring with
1 6= 0.
Definition 1. A ring is called the right (left) Bezout ring if every right
(left) finitely generated ideal is principal. If ring is right and left Bezout
ring at once, it is called Bezout ring
In this work the Bezout domain of the stable range 1 is researched
through the study of influence of the structure of non-principal right
ideals on the diagonal reductions of matrices. It is understandable that
such research narrows the class of the rings which are studied. Further
the Bezout domains that aren’t the rings of the principle right ideals will
be in consideration.
Definition 2. A right ideal of ring, which is maximal in the set of non-
principal right ideals concerning inclusion of right ideals is called maximal
non-principal right ideal. Existing of these right ideals is proved in [9].
Definition 3. A ring R is called a ring with stable range 1, if for any
a, b ∈ R, such that aR + bR = R, exist the element t ∈ R, such that a + bt
is invertible in R [2].
Definition 4. An element a is called right total divisor of element b ∈ R,
(in denotation ar||b,) if there exists duo-element c ∈ R, that bR ⊂ Rc ⊂ aR.
An element c ∈ R is called duo-element of ring R if cR = Rc[3].
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232 Reduction of matrices over Bezout domains
Definition 5. A matrix A over ring R is equivalent to matrix B, (in de-
notation A ∼ B), if there exist invertible matrices P and Q corresponding
sides such that PAQ = B [5].
Definition 6. Let say that in ring R Dubrovin’s condition is true if for
any element a ∈ R there exist duo-element c ∈ R, such that RaR = cR =
Rc [8].
2. Main Results
Proposition 1. Let R is a Bezout domain with Dubrovin’s condition. If
every maximal non- principal right ideal of R is two-sided and element a
doesn’t belong to any one maximal non- principal, right ideal, then every
2 × 2 - matrix A, which element is a, is equivalent to diagonal
(
α 0
0 β
)
,
where αr||β
Proof. According to [9] and limitations imposed on the ring R, we can
state that the matrix A looks like:
(
a 0
b c
)
.
Let’s denote as X a set of divisors x of element a, such that (x, 0) is
the first row of matrices equivalent to A. Since x is a divisor (right or
left) of element a, than any element of X doesn’t belong to any maximal
non-principal right ideal[1]. Let a1 be a minimal element relatively right
division that’s if a1 = p2a2q2 for some a2 ∈ X, then element q2 ∈ U(R).
We need to prove that such element exists.
Let
a = p1a1q1 = p1p2a2q2q1 = . . .
and
aR ⊂ p1a1R ⊂ p1p2a2R ⊂ . . . (1)
Since the element a doesn’t belong to any maximal non-principal right
ideal, then right ideal
∞
⋃
i=1
(
i
∏
j=1
pj)aiR
is principal. Let
∞
⋃
i=1
(
∏i
j=1
pj)aiR = yR.
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T. Kysil, B. Zabavskiy, O. Domsha 233
As y ∈ (
∏i
j=1
pj)aiR, then yR = (
∏i
j=1
pj)aiR. Thus chain (1) finishes
after finitely step.
So, A ∼
(
a1 0
b1 c1
)
, where a1 doesn’t belong to any maximal non-
principal right ideal and a1 ∈ X is minimal relatively the right division.
Let Ra1 + Rb1 = Ra2. According to [7], since R - Bezout domain,
there exists such invertible matrix P1 ∈ GL2(R), that
P1
(
a1 0
b1 c1
)
=
(
a2 b2
0 c2
)
,
where a2 = pa1 for some p ∈ R. As any maximal non-principal right
ideal is two-sided and the element a doesn’t belong to any maximal
non-principal right ideal, a2 also.
Analogically,
(
a2 b2
0 c2
)
Q1 =
(
a3 0
b3 c3
)
for some invertible 2 × 2 matrix Q1 ∈ GL2(R) and a2R + b2R = a3R,
a2 = a3q, q ∈ R. Then a1 = pa3q. As a1 is the minimal element, q ∈ U(R),
that’s why b2R ∈ a2R. In other words, b2 = a2z for some z ∈ R. So,
(
a2 b2
0 c2
)(
1 −z
0 1
)
=
(
a2 0
0 c2
)
.
Let t belongs to R. To the first row of the matrix
(
a2 0
0 c2
)
we shall
add the second, which is multiplied by t. Thinking in the same way we shall
get a2R + tc2R = a1R, whence tc2R ∈ a1R. On the strength of accident
of the element t we can consider that Rc2 ⊂ a2R. So, c2R ⊆ Rc2R ⊆ a2R.
Since the Dubrovin’s condition is true in ring R, then exists such duo-
element c ∈ R, that Rc2R = cR = Rc, what proves the given part of
proposition.
Now we shall consider event, when the element a is located on casual
location of the matrix A. Let’s move it on the location (1,1) by transposi-
tion of the rows and columns of the matrix. So, we can consider that the
matrix A looks like:
A =
(
a b
c d
)
.
Let aR + bR = a′R. According to the definition of the element a
and since a′ the left division of a, follows that a′ doesn’t belong to any
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234 Reduction of matrices over Bezout domains
maximal non-principal right ideal neither. Since R is the Bezout domain,
the matrix A is equivalent to
(
a′ 0
b′ c′
)
[7]. Considering what is proved
above the matrix
(
a′ 0
b′ c′
)
is equivalent to the diagonal matrix with the
condition of the right total division of diagonal elements.
Theorem 1. Let R be the Bezout domain of the stable range 1 with
Dubrovin’s condition. If every maximal non-principal right ideal in R is
two-sided, then any matrix over R is equivalent to the diagonal matrix
with the condition of the right total division of diagonal elements.
Proof. According to [1, 4] for proof of the theorem it is enough to show
that any matrix A =
(
a 0
b c
)
, where RaR+RbR+RcR = R, is equivalent
to the diagonal matrix with the condition of the right total division of
diagonal elements.
Since R is the Bezout domain of the stable range 1, considering [6]
for any a, b ∈ R exist such elements x, d ∈ R, that xa + b = d and
Ra + Rb = Rd. Therefore
(
x 1
1 0
)(
a 0
b c
)
=
(
xa + b c
a 0
)
=
(
d c
a 0
)
,
where a = a0d for some element a0 ∈ R.
Analogically, let cR + dR = zR and cy + d = z for some element y
from R. Then
(
d c
a 0
)(
1 0
y 1
)
=
(
d + cy c
a 0
)
=
(
z c
a 0
)
,
where d = xt, c = zc0 for some elements t, c0 ∈ R. Thereby we proved the
matrix A =
(
a 0
b c
)
is equivalent to the matrix
(
z c
a 0
)
, where a = a0zt
and c = zc0. Since a = a0z, c = zc0 and RaR + RbR + RcR = RzR, then
RaR + RbR + RcR = R, and so RzR = R.
The element z cannot belong to any maximal non-principal right ideal,
therefore according to proposition 1 the matrix
(
z c
a 0
)
, as well as the
matrix
(
a 0
b c
)
is equivalent to the diagonal matrix with the condition of
the right total division of diagonal elements.
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T. Kysil, B. Zabavskiy, O. Domsha 235
References
[1] Kaplansky I. Elementary divisors and modules // Trans. Amer. Math. Soc. – 1949.
– 66. – P. 464 – 491.
[2] Jacobson N. Ring Theory.- M.: Publishers of foreign literature, 1947.
[3] Cohn P.M. Right principal Bezout domains // J. London Math. Soc., 35, N2,
(1987), 251-162.
[4] Zabavskiy B.V., Komarnitskiy N.Ya. Distributive domains jf elementary divisors
// Ukr. Math.J.-1990.-42.-N7,-1000-1004.
[5] Vaserstein L.N. Bass’s first stable range condition // J. of Pure and Appl.
Alg.,1984,-34,319-330.
[6] Zabavskiy B.V. Reduction of matrices over Bezout rings of stable range at most 2
// Ukr. Math.J.,-2003.-55.-N4.-550-554.
[7] Zabavskiy B. V. On noncommutative rings with elementary divisors // Ukr.Math.J.
-1990. -42. -N6. -847-850.
[8] Dubrovin N.I. On noncommutative rings with elementary divisors // Reports of
institutes of higher education. Mathematics,-1986.-N11,-14-20.
[9] Zabavskiy B. V. Factorial analogue of distributive Bezout domains // Ukr.Math.J.-
2001.-53.-N11,-1564-1567.
[10] Amitsur S.A. Remarks of principal ideal rings // Osaka Math.Journ. -1963, -15,
-59-69.
Contact information
T. Kysil Khmelnitsky national university,
The faculty of Applied Mathematics
and Computer Technologies, Applied Mathemat-
ics and Social Informatics Department
E-Mail: kysil_tanya@mail.ru
B. Zabavskiy Lviv national university named after I. Franko,
The faculty of Mechanics and Mathematics,
The chair of Algebra and Logic
E-Mail: b_zabava@franko.lviv.ua
O. Domsha Lviv national university named after I. Franko,
The faculty of Mechanics and Mathematics,
The chair of Algebra and Logic
E-Mail: olya.domsha@i.ua
Received by the editors: 21.04.2012
and in final form 19.05.2012.
|
| id | nasplib_isofts_kiev_ua-123456789-152240 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T17:56:47Z |
| publishDate | 2012 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Kysil, T. Zabavskiy, B. Domsha, O. 2019-06-09T06:08:28Z 2019-06-09T06:08:28Z 2012 Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides / T. Kysil, B. Zabavskiy, O. Domsha // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 230–235. — Бібліогр.: 10 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/152240 It is proved that each matrix over Bezout domain of stable range 1 with Dubrovin's condition, in which every maximal nonprincipal ideals are tho-sides ideals, is equivalent to diagonal one with right total division of diagonal elements. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides Article published earlier |
| spellingShingle | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides Kysil, T. Zabavskiy, B. Domsha, O. |
| title | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides |
| title_full | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides |
| title_fullStr | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides |
| title_full_unstemmed | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides |
| title_short | Reduction of matrices over Bezout domains of stable range 1 with Dubrovin’s condition in which maximal nonprincipal ideals are two-sides |
| title_sort | reduction of matrices over bezout domains of stable range 1 with dubrovin’s condition in which maximal nonprincipal ideals are two-sides |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152240 |
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