Quasi-Euclidean duo rings with elementary reduction of matrices
We establish necessary and sufficient conditions under which a class of quasi-Euclidean duo rings coincides with a class of rings with elementary reduction of matrices. We prove that a Bezout duo ring with stable range 1 is a ring with elementary reduction of matrices. It is proved that a semiexchan...
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| Zitieren: | Quasi-Euclidean duo rings with elementary reduction of matrices / O. Romaniv, A. Sagan // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 317-324. — Бібліогр.: 15 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1551732025-02-09T22:01:02Z Quasi-Euclidean duo rings with elementary reduction of matrices Romaniv, O. Sagan, A. We establish necessary and sufficient conditions under which a class of quasi-Euclidean duo rings coincides with a class of rings with elementary reduction of matrices. We prove that a Bezout duo ring with stable range 1 is a ring with elementary reduction of matrices. It is proved that a semiexchange quasi-duo Bezout ring is a ring with elementary reduction of matrices iff it is a duo ring. 2015 Article Quasi-Euclidean duo rings with elementary reduction of matrices / O. Romaniv, A. Sagan // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 317-324. — Бібліогр.: 15 назв. — англ. 1726-3255 2010 MSC:13F99. https://nasplib.isofts.kiev.ua/handle/123456789/155173 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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We establish necessary and sufficient conditions under which a class of quasi-Euclidean duo rings coincides with a class of rings with elementary reduction of matrices. We prove that a Bezout duo ring with stable range 1 is a ring with elementary reduction of matrices. It is proved that a semiexchange quasi-duo Bezout ring is a ring with elementary reduction of matrices iff it is a duo ring. |
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Romaniv, O. Sagan, A. |
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Romaniv, O. Sagan, A. Quasi-Euclidean duo rings with elementary reduction of matrices Algebra and Discrete Mathematics |
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Romaniv, O. Sagan, A. |
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Romaniv, O. |
| title |
Quasi-Euclidean duo rings with elementary reduction of matrices |
| title_short |
Quasi-Euclidean duo rings with elementary reduction of matrices |
| title_full |
Quasi-Euclidean duo rings with elementary reduction of matrices |
| title_fullStr |
Quasi-Euclidean duo rings with elementary reduction of matrices |
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Quasi-Euclidean duo rings with elementary reduction of matrices |
| title_sort |
quasi-euclidean duo rings with elementary reduction of matrices |
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Інститут прикладної математики і механіки НАН України |
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2015 |
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https://nasplib.isofts.kiev.ua/handle/123456789/155173 |
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Quasi-Euclidean duo rings with elementary reduction of matrices / O. Romaniv, A. Sagan // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 317-324. — Бібліогр.: 15 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT romanivo quasieuclideanduoringswithelementaryreductionofmatrices AT sagana quasieuclideanduoringswithelementaryreductionofmatrices |
| first_indexed |
2025-12-01T05:18:40Z |
| last_indexed |
2025-12-01T05:18:40Z |
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1850281904265232384 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 2, pp. 317–324
© Journal “Algebra and Discrete Mathematics”
Quasi-Euclidean duo rings
with elementary reduction of matrices
O. Romaniv, A. Sagan
Communicated by V. V. Kirichenko
Abstract. We establish necessary and sufficient conditions
under which a class of quasi-Euclidean duo rings coincides with a
class of rings with elementary reduction of matrices. We prove that
a Bezout duo ring with stable range 1 is a ring with elementary
reduction of matrices. It is proved that a semiexchange quasi-duo
Bezout ring is a ring with elementary reduction of matrices iff it is
a duo ring.
Introduction
The problem of the factorization of square matrices over rings was
considered in the mid 1960’s and was formulated in this way: (P) to
characterize the integral domain R, under which an arbitrary invertible
square matrix is a product of elementary matrices. An elementary matrix
with elements of the ring R is understood as a square matrix of one of
the following types:
(1) a diagonal matrix with invertible elements on the main diagonal;
(2) a matrix that differs from the unit matrix by the presence of any
nonzero element outside the main diagonal.
If R is a field, according to the Gauss approach, arbitrary invertible
matrix under it may be decomposed into the product of elementary ma-
trices and the structure of the general linear group GLn(R) is thoroughly
2010 MSC: 13F99.
Key words and phrases: Bezout ring, duo ring, stable range, semiexchange ring,
ring with elementary reduction of matrices.
318 Quasi-Euclidean duo rings
studied (see [5]). The investigation of the integral domain (particularly
non-commutative) that satisfies the conditions of the problem (P), started
in 1966 with the Cohn’s fundamental work [3], who defined these domains
as the general Euclidean (GE-rings, for shortness), due to the fact that
Euclidean domains were the first well-known examples of GE-rings and
are not the fields. Cohn’s work became the reason of the thorough and
detailed study of the general and special linear group’s structure under
different rings. In 1996 Zabavsky B. V. [13] analyzed the rings with the
elementary reduction of matrices and set up a problem of investigation of
such rings.
A ring R is called a ring with elementary reduction of matrices [13]
in case of an arbitrary matrix over R possesses elementary reduction, i.e.
for an arbitrary matrix A over the ring R there exist such elementary
matrices over R , P1, . . . , Pk, Q1, . . . , Qs of respectful size that
P1 · · · Pk · A · Q1 · · · Qs = diag(ε1, . . . , εr, 0, . . . , 0), (1)
where Rεi+1R ⊆ Rεi ∩ εiR for any i = 1, . . . , r − 1.
Since in 1949 Kaplansky [7] established the investigation of the elemen-
tary divisors rings (i. e. the rings under which arbitrary matrix resolves
itself to the accepted diagonality (1) into invertible matrices of the ap-
propriate sizes), so the problem of finding the necessary and sufficient
conditions, whereby a given ring is a ring with elementary reduction of
matrices is closely related to the problem of arbitrary square invertible
matrices decomposition into the product of elementary ones.
Main results
A ring R is understood as an associative ring with nonzero unit element
and U(R) is understood as the group of invertible elements of a ring R.
A group generated by elementary matrices of type (2) of order n is called
a group of elementary matrices En(R), while GEn(R) is understood as a
group of elementary matrices of n order over R.
A right (left) Bezout ring is a ring in which every finitely generated
right (left) ideal is principal. A Bezout ring [6] is a ring which is both
right and left Bezout ring. A ring R is called right Hermite if, for any row
(a, b), a, b ∈ R, there exists an invertible matrix P of order 2 over R so
that (a, b)P = (d, 0), where d ∈ R. Left Hermite rings can be defined by
analogy. If the ring is left and right Hermite, then it is called Hermite
ring [7]. A ring is said to be a right (left) duo ring if any right (left) ideal
O. Romaniv, A. Sagan 319
of this ring is a 2-sided ideal. If the ring is both left and right duo ring,
then it is called duo ring [4].
A ring R is called a right (left) quasi-duo ring, if any right (left)
maximal ideal in R is a two-sided ideal. If the ring is both left and right
quasi-duo ring, then it is called quasi-duo ring [11].
We say that a ring R has quasi-algorithm, if the function ϕ : R × R →
W (where W is some ordinal) is given so that for any a, b ∈ R (b 6= 0)
one can find elements q, r ∈ R such as a = bq + r and ϕ(b, r) < ϕ(a, b).
If one can find some quasi-algorithm on R then the ring R is called
quasi-Euclidean [1].
A ring R is said to have stable range 1, if for any a, b ∈ R satisfying
aR + bR = R, there exists such t ∈ R that a + bt is an invertible element
in R [12]. A ring R is said to have idempotent stable range 1, if for any
a, b ∈ R satisfying aR + bR = R, there exists such idempotent e ∈ R that
a + be is invertible [2].
A ring R is called an exchange ring if for any element a ∈ R there
exists an idempotent e ∈ R such that e ∈ aR and 1 − e ∈ (1 − a)R [9].
Proposition 1. A right quasi-Euclidean ring is right Hermite ring.
Proof. By Theorem 8 [1] for any elements a, b ∈ R, a 6= 0, there exists a
finite divisible chain, that
b = aq1 + r1, a = r1q2 + r2, . . . , rn−2 = rn−1qn + rn, rn−1 = rnqn+1.
Then
(a, b)
(
1 −q1
0 1
)(
1 0
−q2 1
)(
1 −q3
0 1
)
. . .
(
1 −qn+1
0 1
)
= (rn, 0).
So for any elements a, b ∈ R there exists matrix P ∈ GE2(R), that
(a, b)P = (rn, 0). Therefore, R is right Hermite ring.
Lemma 1. Let R be a duo ring. Then for any matrix E ∈ En(R), there
exists such matrix E′ ∈ En(R), that
diag(d, . . . , d) · E = E′ · diag(d, . . . , d).
Proof. The proof follows from the fact, that if R is a duo ring, for any
element a ∈ R, there exists such an element a′ ∈ R, that da = a′d.
320 Quasi-Euclidean duo rings
Proposition 2. A quasi-Euclidean duo ring R is a ring with elementary
reduction of matrices if and only if a matrix of the form
(
a 0
b c
)
∈ M2(R),
where aR + bR + cR = R admits elementary reduction.
Proof. The necessity is obvious. To prove the sufficiency, we consider the
case where aR + bR + cR = dR, d /∈ U(R). By virtue of Proposition 1,
there exist such elements a1, b1, c1 ∈ R, that
a = da1, b = db1, c = dc1 and a1R + b1R + c1R = R.
Then
(
a 0
c b
)
=
(
d 0
0 d
)(
a1 0
b1 c1
)
.
Since the matrix A =
(
a1 0
b1 c1
)
admits elementary reduction, there ex-
ist such elementary matrices P1, . . . , Pk, Q1, . . . , Qs ∈ E2(R) of respectful
size, that
P1 · · · Pk · A · Q1 · · · Qs = diag(ε1, ε2), (2)
where ε1R ∩ Rε1 ⊇ Rε2R.
Multiply equation (2) by the matrix diag(d, d) we obtain:
diag(d, d) · P1 · · · Pk · A · Q1 · · · Qs = diag(dε1, dε2).
According to Lemma 1 there exist such matrices P ′
1, . . . , P ′
k
∈ E2(R), that
P ′
1 · · · P ′
k · diag(d, d) · A · Q1 · · · Qs = diag(dε1, dε2).
Therefore, the matrix
(
a 0
b c
)
also admits elementary reduction.
The proof is completed by induction of the order of matrices.
Theorem 1. Let R is quasi-Euclidean duo ring in which any noninvertible
element belongs to most countable set of maximal ideals of R. Then R is
a ring with elementary reduction of matrices.
According to the Proposition 2, the proof of this theorem repeats the
proof given in [14] in the case of commutative rings, therefore we do not
give it.
O. Romaniv, A. Sagan 321
Corollary 1. A quasi-Euclidean duo ring in which the set of maximal
ideals is at most countable is a ring with elementary reduction of matrices.
Theorem 2. A Bezout duo ring with stable range 1 is a ring with ele-
mentary reduction of matrices.
Proof. By Theorem 2 [15] ring R is a right Hermite ring. It remains
to be proven that the ring R is a ring with elementary reduction of
matrices. According to the Proposition 1, it is sufficient to prove theorem
for matrices
A =
(
a 0
b c
)
∈ M2(R),
where aR + bR + cR = dR for any element d ∈ R. Obviously, there exist
such elements a1, b1, c1 ∈ R, that
a = da1, b = db1, c = dc1 and a1R + b1R + c1R = R.
Then
(
a 0
b c
)
=
(
d 0
0 d
)(
a1 0
b1 c1
)
.
Since R is a right Bezout ring of stable range 1, then for elements
a1, b1, c1 ∈ R there exists such elements s, t ∈ R, that a1s + b1 + c1t =
u ∈ U(R). Multiplying the last equality from the left side on the element
d we get that da1s + db1 + dc1t = as + b + ct = du. Since R is a duo ring,
there exists such element s′ ∈ R, that as = s′a, then s
′
a + b + ct = du.
Considering the matrices P1, P2, P3 ∈ GE2(R)
P1 =
(
1 s′
0 1
)
, P2 =
(
0 1
1 0
)
, P3 =
(
1 0
t 1
)
.
We have
P1P2AP3 =
(
1 s′
0 1
)(
0 1
1 0
)(
d 0
0 d
)(
a1 0
b1 c1
)(
1 0
t 1
)
=
=
(
1 s′
0 1
)(
b c
a 0
)(
1 0
t 1
)
=
(
s′a + b + ct c
a 0
)
=
(
du dc1
da1 0
)
= B
Then the matrix B and, hence, the matrix A obviously admits elementary
reduction. Therefore, R is a ring with elementary reduction of matrices.
322 Quasi-Euclidean duo rings
Corollary 2. A semilocal quasi-Euclidean duo ring is a ring with ele-
mentary reduction of matrices.
Theorem 3. Let R is Hermite duo ring and, for any a, b ∈ R (b 6= 0),
there exists such s ∈ R, that mspec(s) = mspec(a)\mspec(b). Then R is
a ring with elementary reduction of matrices.
Proof. Let a, b ∈ R be such elements, that aR + bR = dR, where d ∈ R.
Note that the case d /∈ U(R) is irrelevant. Otherwise, there exist elements
a1, b1 ∈ R, such that a = da1, b = db1 and a1R + b1R = R. As a result,
we obtain
(
a
b
)
=
(
d 0
0 d
)(
a1
b1
)
. By the fact that R is a duo ring, which
would also imply the existence of mutually prime elements, thus there
exists such a′
1, b′
1 ∈ R, that
(a, b) = (a′
1, b′
1)
(
d 0
0 d
)
.
Therefore, it is sufficient to prove the statement of the theorem for mutually
prime elements. Thus, assume that aR + bR = R. It is obvious that
mspec(a) ∩ mspec(b) = {0}. (3)
Using the statement of the theorem, there exists element s ∈ R, which
belongs to all maximal ideals of the ring R, except for maximal ideals of
the set mspec(a), that is, we have the following equality
mspec(s) = mspec(0)\mspec(a).
It is obvious that
mspec(s) ∩ mspec(a) = {0}. (4)
Let us consider the element a + bs ∈ R and assume that a + bs ∈ M,
where M is a maximal ideal of the ring R. There are the following possible
cases:
1) a ∈ M and b ∈ M contradicts with the condition (3).
2) a ∈ M and s ∈ M contradicts with the condition (4).
Therefore, our initial assumption was incorrect and the condition
a + bs ∈ U(R) is valid. It also implies that R is a ring of a stable range 1.
Due to Theorem 2 a duo ring R is a ring with an elementary reduction of
matrices.
O. Romaniv, A. Sagan 323
We will denote the Jacobson radical of a ring R by J(R). A ring R is
said to be a semiexchange ring [8] if the factor ring R/J(R) is an exchange
ring.
Theorem 4. A semiexchange Bezout duo ring is a ring with elementary
reduction of matrices.
Proof. Let R be a semiexchange Bezout duo ring. Since all idempotent
elements of a duo ring belong to its center, due to Theorem 12 [2], then
R̄ = R/J(R) is a ring with idempotent stable range 1. Since a stable
range 1 lifts modulo J(R), we obtain the result that a ring R also has a
stable range 1. Then, according to Theorem 2, R is a ring with elementary
reduction of matrices.
Theorem 5. Let R be semiexchange quasi-duo Bezout ring. Then R is a
ring with elementary reduction of matrices if and only if it is a duo ring.
Proof. As it was mentioned at the beginning and is proven in [10] being
a quasi-duo elementary divisor ring implies the duo ring condition, so the
necessity is proven.
Sufficiency follows from Theorem 4.
Corollary 3. A distributive semiexchange Bezout ring is a ring with
elementary reduction of matrices if and only if it is a duo ring.
References
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324 Quasi-Euclidean duo rings
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[14] B. V. Zabavsky, O. M. Romaniv Rings with elementary reduction of matrices. Ukr.
mat. J., 52(12)(2000) pp.1641–1649.
[15] B. V. Zabavsky Reduction of matrices over Bezout rings of stable rank not higher
than 2. Ukr. mat. J., 55(4)(2003) pp.665–670.
Contact information
O. M. Romaniv Department of Mechanics and Mathematics,
Ivan Franko National Univ., Lviv, Ukraine
E-Mail(s): oromaniv@franko.lviv.ua
A. V. Sagan Department of Mechanics and Mathematics,
Ivan Franko National Univ., Lviv, Ukraine
E-Mail(s): andrijsagan@gmail.com
Received by the editors: 17.11.2015
and in final form 17.11.2015.
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