Semiperfect ipri-rings and right Bézout rings
We present a survey of some results on ipri-rings and right Bézout rings. All these rings are generalizations of principal ideal rings. From the general point of view, decomposition theorems are proved for semiperfect ipri-rings and right Bézout rings.
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| author | Gubareni, N. M. Dokuchaev, M. A. Kirichenko, V. V. Губарені, Н. М. Докучаєв, М. А. Кириченко, В. В. |
| author_facet | Gubareni, N. M. Dokuchaev, M. A. Kirichenko, V. V. Губарені, Н. М. Докучаєв, М. А. Кириченко, В. В. |
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| description | We present a survey of some results on ipri-rings and right Bézout rings. All these rings are generalizations of principal ideal rings. From the general point of view, decomposition theorems are proved for semiperfect ipri-rings and right Bézout rings. |
| first_indexed | 2026-03-24T02:32:22Z |
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| fulltext |
UDC 512.552.12
M. A. Dokuchaev (Univ. São Paulo, Brazil),
N. M. Gubareni (Techn. Univ. Czȩstochowa, Poland),
V. V. Kirichenko (Kiev Nat. Taras Shevchenko Univ., Ukraine)
SEMIPERFECT IPRI-RINGS AND RIGHT BÉZOUT RINGS*
НАПIВДОСКОНАЛI IПРI-КIЛЬЦЯ ТА ПРАВI КIЛЬЦЯ БЕЗУ
We present a survey of some results on ipri-rings and right Bézout rings. All these rings are generalizations of
principal ideal rings. From the general point of view, decomposition theorems for semiperfect ipri-rings and
for right Bézout rings are proved.
Наведено огляд результатiв з теорiї iпрi-кiлець та правих кiлець Безу. Цi кiльця є узагальненням кiлець
головних iдеалiв. Iз загальної точки зору доведено теореми розкладу для напiвдосконалих iпрi-кiлець
та правих кiлець Безу.
1. Introduction. Recall that a principal right ideal ring is a ring with identity 1 6= 0 in
which every right ideal is principal (A. W. Goldie [8] called it a pri-ring). A principal left
ideal ring (a pli-ring) can be defined analogously. Properties of pli-rings were considered
in [15].
A principal ideal ring is a ring which is both a principal right and principal left ideal
ring. A ring A with the Jacobson radical R is a primary ring if A/R is a simple Artinian
ring.
One of the main examples of a principal ideal ring is the ring Z of all integers.
It is well-known that every commutative principal ideal ring is a finite direct sum of
rings which are either integral domains or are completely primary (see [33], Chapter 4).
Analogous theorem was proved by K. Asano for the case of non-commutative Artinian
rings. He proved in [1] that each such a ring is a finite direct sum of primary rings.
A. W. Goldie considered the structure of pri-rings. He proved in [8] the following
main theorems.
Theorem A. A pri-ring with no nilpotent ideals is a finite direct sum of prime
pri-rings.
Theorem B. A prime pri-ring is a complete matrix ring Kn, where K is a right
Noetherian integral domain.
Theorem C. A pri-ring, which is left Noetherian, is a finite direct sum of pri-rings,
each being either a prime ring or a primary ring.
J. C. Robson [25] considered a wider class of pri-rings which he called ipri-rings and
ipli-rings. An ipri-ring (ipli-ring) is a ring in which every two-sided ideal is a principal
right (left) ideal. A ring A with the nilpotent radical W is called W-simple if A/W
is a simple ring. J.C. Robson extended several results concerning pri-rings which were
proved by A.W. Goldie. In particular, he proved the following theorems.
Theorem 1 [25]. A Noetherian ipri-ring is a finite direct sum of ideals each of
which is a Noetherian ipri-ring and is either prime or W-simple. A Noetherian ipri-ring
has a (right and left) quotient ring which is an Artinian pri-ring.
Theorem 2 [25]. If A is a Noetherian ipri- and ipli-ring then multiplication of
ideals in A is commutative; A is a direct sum of rings each of which is prime or W-
*This work was partially supported by CNPq and FAPESP of Brazil.
c© M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO, 2010
612 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
SEMIPERFECT IPRI-RINGS AND RIGHT BÉZOUT RINGS 613
simple, and in each of these rings every proper ideal is a unique product of maximal
ideals.
Note that one-sided Artinian ipri- and ipli-rings were considered earlier by N. Jacobson
in [14].
Theorem I ([14], Theorem 37). If A is a ring with an identity satisfying the descen-
ding chain condition on one-sided ideals, and every two-sided ideal of A is a principal
right ideal and a principal left ideal, then A is a direct sum of two-sided ideals which
are primary rings having these properties.
In [9] (§ 12.2) an analogous theorem was proved for semiperfect rings.
Definition 1.1. A ring O (not necessary commutative) is called a principal ideal
domain if it has no zero divisors and all its right and left ideals are principal.
Theorem 1.1 ([9], § 12.2). Let A be a semiperfect ring such that every two-sided
ideal in A is both a right principal ideal and a left principal ideal. Then A is a principal
ideal ring isomorphic to a direct product of a finite number of full matrix rings over
Artinian uniserial rings and local principal ideal domains. Conversely, all such rings
are semiperfect principal ideal rings.
Another generalization of pri-rings are right Bézout rings.
Definition 1.2. A ring is said to be right (resp. left) Bézout ring if its every finitely
generated right (left) ideal is principal. A ring which is a right and left Bézout ring
is called a Bézout ring. A Bézout domain is an integral domain in which every finitely
generated ideal is principal.
A pri-ring is obviously a right Bézout ring. In a certain sense, a right Bézout ring
is a non-Noetherian analog of a pri-ring. On the other hand from the fact that any right
ideal in a right Noetherian ring is finitely generated it immediately follows the following
statement.
Proposition 1.1. A right Noetherian ring is a right Bézout ring if and only if it is
a pri-ring.
The main examples of commutative Bézout domains which are not principal ideal
domains (PID) and not Noetherian are:
1. The ring O(D) of all functions in single complex variable holomorphic in a
domain D of the complex plane C.
2. The ring of holomorphic functions given on the entire complex plane C.
3. The ring of all algebraic integers.
First the properties of the ring O(D) of all functions in single complex variable
holomorphic in a domain D of the complex plane C was considered by J. H. M. Wed-
derburn in 1915 [29]. In this paper he considered the problem of reducing the matrix
whose coefficients are functions from the ring O(D) to an equivalent diagonal matrix.
In particular, he proved the main lemma:
Lemma 1.1 (Wedderburn [29]). Let f, g ∈ O(D) be two holomorphic functions
which are holomorphic in a domain D ⊂ C and which are relatively prime, i.e., have
no common zeros in D. Then there exist two functions p, q ∈ O(D) holomorphic in D
such that
pf + qg = 1.
Wedderburn’s lemma together with Mittag-Leffler series and Weierstrass products
makes it possible to prove that in the ring O(D) any finitely generated ideal is principal.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
614 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Below an example of ideals in O(D) that are not finitely generated is given. This
example and the historical notes about ideal theory in rings of holomorphic functions
can be found in the very interesting book on complex function theory [24].
Let S be an infinite locally finite set in a domain D ⊂ C. The set
I =
{
f ∈ O(D) : f vanishes almost everywhere on S
}
is an ideal in O(D) which is not finitely generated.
From this example the following statement follows.
Proposition 1.2 [24, p. 136]. No ring O(D) is Noetherian; in particular, O(D) is
never a principal ideal ring.
It worth to say that even a great algebraist like J. H. M. Wedderburn in his paper
[29] with famous Lemma 1.1 which is a basis for building the ideal theory of functions
holomorphic in an arbitrary domain D ⊂ C wrote nothing about ideal theory. O. Helmer
was the first who considered the ideal theory of holomorphic functions. In paper [12]
in 1940 he extensively investigated divisibility properties of the ring O(K) = K〈z〉 of
integral functions on K, i.e., functions in single variable z holomorphic on K, where
K is an arbitrary subfield of C. In this paper O. Helmer independently proved the
Wedderburn lemma for D = K. His main result was that any finitely generated ideal
in O(K) is principal. In his next paper [11] O.Helmer studied the algebraic structure
of an abstract commutative ring A in which every finitely generated ideal is principal,
and which further satisfies the following conditions: for any a, c ∈ A with a 6= 0 one
can write a = rs with (r, c) = 1 and (s1, c) 6= 1 for any non-unit divisor s1 of s.
O. Helmer called this ring an adequate ring. The ring of integral functions forms an
excellent example of such a ring. In paper [12], Theorem 8, O. Helmer also gives the
following example of an ideal S in K〈z〉 which is not principal:
S =
(
sin z, sin
1
2
z, sin
1
4
z, . . .
)
.
The ideal structure of the ring of entire functions, also called integral functions,
which are complex-valued functions that is holomorphic over the whole complex plane
C, were studied also by M. Henrikson and L. Gilman (see [6, 7, 13]). The particular
attention in these papers were paid to maximal and prime ideals. An abstract ring which
all finitely generated ideals are principal they called an F-ring.
An example of a non-commutative Bézout ring is the (right) ring of skew formal
series K[[x, σ]], where K is a field, σ is a nontrivial automorphism of K, with multipli-
cation defined by the rule axi = xiσi(a) for any a ∈ K and i ∈ N. Indeed this ring is
a noncommutative pri-ring.
The theory of principal ideal rings and their generalizations has a long and very
interesting history (see [2 – 6, 11, 12, 15, 16, 18, 21, 22, 24 – 32] and many other papers).
The description of left Bézout semiperfect rings was obtained by R. B. Warfield, Jr.
in [27].
Theorem 1.2 (Warfield [27]). A semiperfect ring A is left Bézout if and only if A
is a direct product of a finite number of full matrix rings over left uniserial rings.
For generalizations of this theorem see [28], and see also [9] (Chapter 12) for further
reading and references.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
SEMIPERFECT IPRI-RINGS AND RIGHT BÉZOUT RINGS 615
In this paper we consider semiperfect pri-rings, ipri-rings and right Bézout rings.
The Annihilation lemma (see Lemma 2.1) and Corollary 2.2 from the Generator lemma
(see Lemma 2.2) are used to prove from the general point of view the decomposition
theorems: all these rings are direct products of a finite number of full matrix rings over
uniserial rings. In particular, it is given another proof of the first part of Theorem 1.2.
All rings in this paper are assumed to be associative (but not necessary commutative)
with 1 6= 0 and all modules are assumed to be unitary.
2. Semiperfect rings. Let I be a two-sided ideal of a ring A. One says that
idempotents can be lifted modulo an ideal I of A if from the fact that g2− g ∈ I, where
g ∈ A, it follows that there exists an idempotent e2 = e ∈ A such that e− g ∈ I.
Proposition 2.1 ([9], § 10.3). Idempotents can be lifted modulo any nil-ideal I of a
ring A.
Corollary 2.1 ([9], § 10.3). Idempotents can be lifted modulo the radical of an Arti-
nian ring.
Let radA = R be the Jacobson radical of a ring A. Recall that a ring A is called
semilocal if A/R is a right Artinian ring, or equivalently, if A/R is a semisimple ring.
Any local ring is a semilocal and any right (or left) Artinian ring is semilocal.
LetM be an arbitrary A-module. Denote by radM the intersection of all its maximal
submodules. By convention, ifM does not have maximal submodules, then radM = M.
This submodule is called the radical of the module M.
Proposition 2.2 ([9], § 5.1). Let A be a ring with the Jacobson radical R. If P is a
nonzero projective A-module, then radP = PR 6= P.
If A is a semilocal ring then one has the analogous proposition for any A-module M.
Proposition 2.3 ([20], Proposition 24.4). If M is a nonzero right A-module over a
semilocal ring A then radM = MR.
Recall some definitions and main facts on semiperfect rings (see [9], Chapters 10,
11). Let R be the Jacobson radical of a ring A.
Definition 2.1. A ring A is called semiperfect if A is semilocal and idempotents
can be lifted modulo the Jacobson radical R of A.
Right Artinian rings and local rings are examples of semiperfect rings.
Let A be a ring and e2 = e ∈ A be a nonzero idempotent of A. An idempotent
e ∈ A is called local if the ring eAe is local.
A submodule N of a module M is called small if the equality N +X = M implies
X = M for any submodule X of the module M.
Definition 2.2. A projective module P is called a projective cover of a module M
and it is denoted by P (M) if there is an epimorphism ϕ : P →M such that Kerϕ is a
small submodule in P.
If a simple module U has a projective cover P (U) then P (U) has exactly one
maximal submodule radP (U) and P (U)/ radP (U) ∼= U.
If a module M has a projective cover P (M), then the projective cover is unique up
to isomorphism. The projective cover P (M) of M, where M = M1 ⊕M2, is equal to
P (M1)⊕ P (M2).
The following four theorems are the main theorems in the theory of semiperfect rings
(see [9], Chapter 10).
Theorem 2.1. A ring A is semiperfect if and only if it can be decomposed into a
direct sum of right ideals each of which has exactly one maximal submodule.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
616 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Theorem 2.2 (B. J. Müller). A ring A is semiperfect if and only if 1 ∈ A can be
decomposed into a sum of a finite number of pairwise orthogonal local idempotents.
Theorem 2.3 (H. Bass). The following conditions are equivalent for a ring A :
(a) A is semiperfect;
(b) any finitely generated right A-module has a projective cover;
(c) any cyclic right A-module has a projective cover.
Theorem 2.4. Any indecomposable projective module over a semiperfect ring A is
finitely generated, it is a projective cover of a simpleA-module and has exactly one maxi-
mal submodule. There is a one-to-one correspondence between mutually nonisomorphic
indecomposable projective A-modules P1, . . . , Ps and mutually nonisomorphic simple
A-modules which is given by the following correspondences: Pi 7→ Pi/PiR = Ui and
Ui 7→ P (Ui).
Definition 2.3. An indecomposable projective right module over a semiperfect
ring A is called a principal right module. A principal left module is an indecomposable
projective left A-module.
Note that any principal right (resp. left) A-module is exactly a cyclic indecomposable
projective module, and it has the form eA (resp. Ae), where e is a local idempotent.
Write Xn = X ⊕ . . .⊕X︸ ︷︷ ︸
n times
for any right A-module X and X0 = 0.
Let A be a semiperfect ring with the Jacobson radical R.
Proposition 2.4 ([9], Proposition 11.1.1). LetA = Pn1
1 ⊕. . .⊕Pns
s be the decomposi-
tion of a semiperfect ring A into a direct sum of principal right A-modules and let
1 = f1 + . . . + fs be a corresponding decomposition of the identity of A into a sum
of pairwise orthogonal idempotents, i.e., fiA = Pni
i . Then the Jacobson radical of the
ring A has a two-sided Peirce decomposition of the following form:
R =
R11 A12 . . . A1n
A21 R22 . . . A2n
...
...
. . .
...
An1 An2 . . . Rnn
,
where Rii = rad (fiAfi), Aij = fiAfj for i, j = 1, . . . , n.
The ring fiAfi is isomorphic to EndA(Pni
i ) 'Mni
(End(Pi)), where EndA(Pi) =
= O is a local ring by [9] (Theorem 10.3.8). By [9] (Proposition 3.4.10), radMni
(O) =
= Mni
(radOi).
Set Ui = Pi/PiR. Since Ā = A/R = Un1
1 ⊕ . . .⊕ Uns
s , the idempotents f1, . . . , fs
are central modulo the radical and all simple right A-modules are exhausted by the
modules U1, . . . , Us. Analogously, if Vi = Qi/RQi, then all simple left A-modules are
exhausted by the modules V1, . . . , Vs.
Lemma 2.1 (Annihilation lemma). Let 1 = f1+. . .+fs be a canonical decomposi-
tion of the identity of a semiperfect ring A. For every simple right A-module Ui and for
each fj , Uifj = δijUi, i, j = 1, . . . , s. Similarly, for every simple left A-module Vi and
for each fj , fjVi = δijVi, i, j = 1, . . . , s.
The following lemma allows to compute the minimal number of generators µA(X)
of a finitely generated module X over a semiperfect ring A.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
SEMIPERFECT IPRI-RINGS AND RIGHT BÉZOUT RINGS 617
Lemma 2.2 (Generator lemma). Let A =
s
⊕
i=1
Pni
i be the decomposition of a semi-
perfect ring A into a direct sum of principal right A-modules, X be a finitely generated
right A-module and P (X) =
s
⊕
i=1
Pmi
i be a projective cover of X. If m = max mi
ni is
an integer, then µA(X) = m. Otherwise, µA(X) = [m] + 1.
Here [m] is the integral part of m, i.e., the largest integer ≤ m.
At first this lemma was proved in [17], see also [9] (Lemma 11.1.8). From this lemma
it immediately results the following main corollary.
Corollary 2.2. Let M be a finitely generated right A-module and P (M) =
= Pm1
1 ⊕ . . . ⊕ Pms
s be a projective cover of M. A module M is cyclic if and only if
m1 ≤ n1, . . . ,ms ≤ ns.
Note that this corollary is the same as Lemma 1.8 in [27].
Recall definitions of serial modules and rings (see [9], § 12.1).
Definition 2.4. A module is called uniserial if the lattice of its submodules is a
chain, i.e., the set of all its submodules is linearly ordered by inclusion. A module is
called serial if it decomposes into a direct sum of uniserial submodules.
Definition 2.5. A ring is called right (resp. left) uniserial if it is a right (resp. left)
uniserial module over itself, i.e., the lattice of right ideals is linearly ordered. A ring is
called right (resp. left) serial if it is a right (resp. left) serial module over itself. A ring
which is both a right and left serial ring is called a serial ring.
Note that a right uniserial ring is local, and a right serial ring is semiperfect.
For a notion of a quiver Q(A) of a semiperfect right Noetherian ring A see [9]
(Chapter 11).
Let Q = (V Q,AQ, s, e) be a quiver (directed graph, or digrapf), which is given by
two sets V Q, AQ and two mappings s, e : AQ→ V Q. The elements of V Q are called
vertices or points, and those of AQ arrows. Usually the vertices of Q will be denoted by
numbers 1, 2, . . . , s. If an arrow σ ∈ AQ connects the vertex i ∈ V Q with the vertex
j ∈ V Q, then i = s(σ) is called its start vertex (or source vertex) and j = e(σ) is called
its end vertex (or target vertex). This will be denoted as σ : s(σ) → e(σ), or shortly
σ : i→ j.
A path of a quiver Q from the vertex i to the vertex j is an ordered set of k arrows
{σ1, σ2, . . . , σk} such that the start vertex of each arrow σm coincides with the end
vertex of the previous one σm−1 for 1 < m ≤ k, and moreover, the vertex i is the start
vertex of σ1, while the vertex j is the end vertex of σk. The number k of arrows is
called the length of the path.
The start vertex i of the arrow σ1 is called the start of the path and the end j of the
arrow σk is called the end of the path. The path is said to connect the vertex i with the
vertex j and this is denoted by σ1σ2 . . . σk : i→ j.
By convention it is considered that the path εi of length zero connects vertex i with
itself without any arrow.
Definition 2.6. A path, connecting a vertex of a quiver with itself and of length
not equal to zero, is called an oriented cycle. An oriented cycle of the length 1 is called
a one-pointed cycle or a loop. A quiver without multiple arrows and multiple loops is
called a simply laced quiver.
Remark 2.1. Note that first the notion of quiver was introduced by P. Gabriel for
finite directed graphs when both sets V Q and AQ are finite.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
618 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Definition 2.7. A serial ring is called a primary decomposable serial ring if it is
isomorphic to a finite direct product of primary rings.
Theorem 2.5 ([9], § 12.4). For a semiperfect two-sided Noetherian ring A the fol-
lowing conditions are equivalent:
(a) A is a principal ideal ring;
(b) A is a primary decomposable serial ring;
(c) both the right and left quiver of A is a disconnected union of points and one-
pointed cycles.
Recall that a ring A is called quasi-Frobenius if it is a self-injective two-sided
Artinian ring.
Theorem 2.6. The following conditions are equivalent for a two-sided Artinian
indecomposable ring A.
1. A is a principal ideal ring.
2. A is a quasi-Frobenius ring and A has a minimal (by inclusion) two-sided ideal
I such that A/I is quasi-Frobenius.
3. A ring A and all its factor rings A/I for each ideal I of A are quasi-Frobenius.
Proof. 1⇐⇒ 2. This is Theorem 4.15.5 [10].
3 =⇒ 2. It is trivial.
1 =⇒ 3. By Theorem 2.5, A is a primary serial ring which is a full matrix ring
Mn(B), where B is an Artinian uniserial ring. Let M be the Jacobson radical of B.
Then R = Mn(M) is the Jacobson radical of A and any two-sided ideal of A has the
form Ri = Mn(Mi). Therefore each ring A/Ri is quasi-Frobenius.
3. Decomposition theorems. Right Bézout rings and ipri-rings are natural generali-
zation of pri-rings. Clearly, every pri-ring is a right Noetherian right Bézout ring and a
right Noetherian ipri-ring. In the general case there are two strict inclusions:
pri-rings ⊂ ipri-rings,
pri-rings ⊂ right Bézout rings.
Example 3.1. Let p be a prime integer, Q be the field of rational numbers,
Zp =
{
m
n
∈ Q
∣∣∣∣ (n, p) = 1
}
.
Consider the following ring
O =
{(
α β
0 α
)∣∣∣∣α ∈ Zp, β ∈ Q
}
.
The unique maximal ideal of the ring O is the following ideal
M =
{(
pα β
0 pα
)∣∣∣∣α ∈ Zp, β ∈ Q
}
and
∞⋂
k=1
Mk =
{(
0 β
0 0
)∣∣∣∣β ∈ Q
}
= X,
which is a uniserial A-module.
The ring O is a commutative uniserial ring and therefore O is a right Bézout ring.
But O is neither a pri-ring, nor ipri-ring, since X is a principal ideal.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
SEMIPERFECT IPRI-RINGS AND RIGHT BÉZOUT RINGS 619
In this section decomposition theorems are given under the additional condition that
they are all semiperfect.
Recall that a module M is called local if M has the largest proper submodule.
Equivalently, a module M is local if it is cyclic, non-zero and has the unique maximal
proper submodule.
Proposition 3.1. Let A be a local ring with the unique maximal ideal R. A right
A-module M is local if and only if M = mA for some m ∈M.
Proof. If M is a local module, then M is cyclic, which follows from the definition.
Inversely, assume that M is a cyclic module. Since A is a local ring, from Proposi-
tion 2.3 it follows that radM = MR. Therefore M/MR is a semisimple module.
If M/MR is not simple, then M contains at least two maximal submodules. Then
the projective cover P (M) ∼= Am where m ≥ 2. This implies, by Corollary 2.2, that M
is not cyclic.
ThereforeM/MR is simple, which means thatMR = radM is a maximal submodule
of M. Since M is cyclic, MR is the unique maximal submodule of M, by Nakayama’s
lemma. So that M is a local module.
Lemma 3.1. Let A be a local ring with a unique maximal ideal R. Then any
principal right ideal I of A has a unique maximal submodule which has the form IR.
Moreover, if I = L1 + . . .+Lm, where Li are right ideals of A, then there exists i such
that I = Li.
Proof. Let I be a principal right ideal of A. By Proposition 2.3, IR is the intersection
of all maximal submodules (right ideals) of I. If I contains at least, two maximal
submodules, then the projective cover P (I) ∼= Am where m ≥ 2, and so I is not
principal, by Corollary 2.2. Consequently, a principal right ideal I is a local module.
If I = L1 + . . . + Lm, where Li are right ideals of A, then either Li ⊂ IR, or
Li = I. If all Li ⊂ IR, then I ⊂ IR. It is a contradiction.
Theorem 3.1. A local ipri-ring O is right uniserial.
Proof. LetM be the unique maximal ideal of O. For any n > 0 an idealMn is a
two-sided ideal, and so it is a principal right ideal. By Lemma 3.1,Mn has the unique
maximal submoduleMnM =Mn+1. So one has the chain of principal right ideals
O ⊃M ⊃M2 ⊃ . . . ⊃Mn ⊃ . . .
with simple factors. Since O is an ipri-ring,Mω =
∞⋂
n=1
Mn is a two-sided ideal which
is a principal right ideal.
Let I be a two-sided ideal of a ring A. Define the transfinite powers of I following
to [19]:
I1 = I,
Iβ+1 = Iβ · I,
Iα =
⋂
β<α
Iβ , for any limit ordinal α.
All transfinite powers ofM are two-sided ideals, and, sinceO is an ipri-ring, they are
all principal right ideals. Therefore taking into account this definition and Lemma 3.1,
we obtain the following chain of principal right ideals of O:
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
620 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
O ⊃M ⊃M2 ⊃ . . . ⊃Mn ⊃ . . . ⊃Mω ⊃Mω+1 ⊃Mω+2 ⊃ . . .
with simple factors, which implies that O is a right uniserial ring.
Theorem 3.2. Let A be a semiperfect ipri-ring. Then A is a finite direct product
of full matrix rings over local rings.
Proof. Let A = Pn1
1 ⊕ . . .⊕Pns
s be a decomposition of a semiperfect ring A into a
direct sum of principal A-modules, and 1 = f1 + . . .+ fs be a corresponding canonical
decomposition of the identity of A into a sum of pairwise orthogonal idempotents, i.e.,
fifj = δijfj (i, j = 1, . . . , s) and fiA = Pni
i , i = 1, . . . , s.
Let 1 = h1 + h2, where h1 = f1 and h2 = 1 − f1 are idempotents of A. Then
R =
(
R1 X
Y R2
)
, where A1 = h1Ah1, A2 = h2Ah2, X = h1Ah2 and Y = h1Ah2.
By Proposition 2.4, R =
(
R1 X
Y R2
)
, where Ri is the Jacobson radical of Ai, i = 1, 2.
Consider the following two-sided ideal I of A: I =
(
A1 X
Y Y X
)
. Obviously,
I2 = I and I = gA for some g ∈ I since A is an ipri-ring. Then I = h1gA ⊕ h2gA.
Therefore, h1I = h1gA = (A1, X) and h2I = h2gA = (Y, Y X).
One has
IR =
R1 X
Y R1 Y X
.
Suppose, Y 6= 0. By Nakayama’s lemma, (R1, X) 6= (A1, X) and (Y R1, Y X) 6=
6= (Y, Y X). Consider the semisimple A-module I/IR. We have that (I/IR)f1 =
= I/IR and the Annihilation lemma I/IR = Um1 , where m > n1 (note that Y 6= 0).
Therefore, P (I) = Pm1 . By Corollary 2.2, m ≤ n1. This contradiction shows that
Y = 0 and A =
(
A1 X
0 A2
)
. Obviously, K =
(
0 X
0 A2
)
is a two-sided ideal in A.
Hence, K = h1tA ⊕ h2tA, where K = tA. Let X 6= 0. Then for the semisimple A-
module K/KR we have (K/KR)f1 = 0. Therefore, the Annihilation lemma K/KR =
= A2/R2⊕X/XR2 is a direct sum of simpleA-modules U2, . . . , Us. A moduleX/XR2
is nonzero by Nakayama’s lemma. Let Uk be a direct summand of X/XR2, 2 ≤ k ≤ s.
Then P (K) contains a direct summand Pnk+1
k and by Corollary 2.2 K is not a principal
right ideal. As above X = 0.
Consequently, A = Mn1(EndA P1)×EndA(Pn2
2 ⊕. . .⊕Pns
s ). Applying the inducti-
on by s we obtain EndA(Pn2
2 ⊕ . . .⊕ Pns
s ) = Mn2
(EndA P2)× . . .×Mns
(EndA Ps).
Since A is a semiperfect ring, all rings O1 = EndA P1, O2 = EndA P2, . . . ,Os =
= EndA Ps are local.
The theorem is proved.
From Theorems 3.1 and 3.2 it follows the following main theorem:
Theorem 3.3 (Decomposition theorem for semiperfect ipri-rings). Let A be a semi-
perfect ipri-ring. Then A is a finite direct product of full matrix rings over right uniserial
rings.
Since any pri-ring is a right Noetherian ipri-ring, we immediately have the following
corollary.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
SEMIPERFECT IPRI-RINGS AND RIGHT BÉZOUT RINGS 621
Corollary 3.1. A semiperfect pri-ring is a finite direct product of full matrix rings
over right uniserial rings.
From Theorem 3.3 and Warfield’s Theorem 1.2 there results the following statement.
Corollary 3.2. A semiperfect ipri-ring is a right Bézout ring.
Thus in the case of semiperfect rings one has the following chain of rings:
pri-rings ⊂ ipri-rings ⊂ right Bézout rings.
Lemma 3.2. Let A = Pn1
1 ⊕ . . . ⊕ Pns
s be a decomposition of a semiperfect
ring A into a direct sum of principal right A-modules. Let φ : Pi → Pj be a nonzero
homomorphism (i 6= j). (One can assume that i < j.) The right ideal of the form
Jφ = Pn1
1 ⊕ . . .⊕ P
ni
i ⊕ . . .⊕ P
nj−1
j ⊕ Imφ⊕ Pnj+1
j+1 ⊕ . . .⊕ P
ns
s
is not cyclic.
Proof. Note that Jφ is a finitely generated ideal. Let P (Jφ) be the projective cover
of Jφ. Then P (Jφ) = Pn1
1 ⊕ . . .⊕P
ni+1
i ⊕ . . .⊕Pnj−1
j ⊕ . . .⊕Pns
s . By Corollary 2.2,
Jφ is not cyclic.
Corollary 3.3. Let A be a semiperfect right Bézout ring with decomposition A =
= Pn1
1 ⊕ . . .⊕ Pns
s into a direct sum of principal right A-modules. Then
HomA
Pi,∑
k 6=i
⊕Pnk
k
= 0.
The following theorem is a part of Theorem 1.14 [27] proved by R. B. Warfield, Jr.
Here it is given some other proof of this theorem which shows its close connection with
the proof of Theorem 3.2.
Theorem 3.4 (Decomposition theorem for right Bézout semiperfect rings). A right
Bézout semiperfect ring A is a finite direct product of full matrix rings over local rings.
Proof. Let AA = Pn1
1 ⊕ . . . ⊕ Pns
s be a decomposition as above, and let s ≥ 2.
Let f1A = Pn1
1 , where f21 = f1 and e = 1 − f1. Consider the two-sided Peirce
decomposition of A with respect to the decomposition of 1 = f1 + e:
A =
(
A1 X
Y A2
)
where A1 = f1Af1, A2 = eAe, X = f1Ae and Y = eAf1:
By Corollary 3.3, HomA(Pn1
1 , Pn2
2 ⊕ . . .⊕ Pns
s ) = 0, whence Y = eAf1 = 0.
Suppose X 6= 0 and x ∈ X, x 6= 0. We have x = xf2 + . . . + xfs. So, there
exists an i, such that (2 ≤ i ≤ s) and xfi 6= 0. Therefore, f1xfi defines a nonzero
homomorphism θ : Pni
i → Pn1
1 by the formula
θ(fia) = f1xfia.
By Corollary 3.3, we obtain a contradiction.
Therefore X = 0. Consequently, A = Mn1(EndA P1) × EndA(Pn2
2 ⊕ . . . ⊕ Pns
s ).
Applying the induction by s we have EndA(Pn2
2 ⊕ . . .⊕ Pns
s ) = Mn2
(EndA P2)× . . .
. . .×Mns
(EndA Ps).
The theorem is proved.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
622 M. A. DOKUCHAEV, N. M. GUBARENI, V. V. KIRICHENKO
Theorem 3.5. A local ring O is right Bézout if and only if O is right uniserial.
Proof. Let O be a right uniserial, and let J = x1O + . . . + xnO be a finitely
generated right ideal of O. Assume that x1, . . . , xn is a minimal system of generators
of J and n ≥ 2. Consider the right ideals x1O and x2O. Since O is right uniserial, one
can assume that x1O ⊆ x2O and J = x1O + x3O + . . .+ xnO. Therefore, x1, . . . , xn
is not a minimal system of generators of J. This contradiction shows that n = 1, and so
O is a right Bézout ring.
Conversely, let O be a right Bézout local ring, but O is not a right uniserial ring.
Then there exist right ideals X and Y such that X + Y strongly contains X and Y.
Moreover, X % X ∩ Y , and Y % X ∩ Y. Consequently, there exist x and y such that
x ∈ X and x 6∈ Y and y ∈ Y, y 6∈ X. Denote by K = xO + yO a two-generated right
ideal of O. Since O is right Bézout, K = zO.
By Lemma 3.1, K is a local module, and either K = xO or K = yO. Consequently,
a right Bézout local ring is right uniserial.
The theorem is proved.
From Theorems 3.4 and 3.5 the main decomposition theorem for right Bézout semi-
perfect rings follows.
Theorem 3.6 (Decomposition theorem for right Bézout semiperfect rings). A right
Bézout semiperfect ring A is a finite direct product of full matrix rings over uniserial
rings.
It is easy to see that the properties (1) ipri; (2) pri; (3) right Bézout hold if and only
if these properties hold for finite direct products and direct summands of rings with these
properties.
Proposition 3.2. Let O be a local ring. Then Mn(O) is a right Bézout ring if and
only if O is a right Bézout ring.
Proof. Let A = Mn(O) be a right Bézout ring with the maximal ideal M, then
radA = Mn(M). Suppose that L is a finitely generated right ideal of O. Then L̃ =
= (L, . . . , L)︸ ︷︷ ︸
n
is a finitely generated right ideal ofMn(O). Therefore L̃ is a principal right
ideal and L̃/L̃R = (L/M, . . . , L/M). By Corollary 2.2, L/M is a simple module.
Hence, L is a principal right ideal.
A converse statement follows from Lemma 1 [28].
The proposition is proved.
Let A be a finite ipri-ring. If A is an indecomposable ring then, by Theorem 3.2,
A = Mn(O), where O is a finite right uniserial ring.
Let O be a local ring with the unique (right, left and two-sided) ideal M and Ō =
= O/M. The module Ō is the unique right (resp., left) simple O-module U (resp., V ).
Denote the number of elements in a finite set S by |S|.
Proposition 3.3. Let O be a finite local ring withM2 = 0 (M 6= 0), Ō = O/M
andM as a right O-module is isomorphic to U t. Then |O| = |U |t+1.
Proof. Let |U | = m. Then, by the Lagrange theorem, |O| = [O : M] · |M| =
= m · |U |t = |U |t+1.
Corollary 3.4. Let O and M be as in Proposition 3.3. Then M as a left O is
isomorphic to V t.
The proof follows from the previous proposition taking into account that |U | = |V |.
Proposition 3.4. Right and left quivers of a finite local ring coincide.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
SEMIPERFECT IPRI-RINGS AND RIGHT BÉZOUT RINGS 623
The proof follows from Corollary 3.4 and the definition of a quiver.
Proposition 3.5. A right Artinian ring A with the radical R is right uniserial if
and only if A/R2 is a right uniserial ring.
The proof follows immediately from [9] (Theorem 12.3.10).
Note, that a right uniserial ring O is local.
Proposition 3.6. A right uniserial finite ring O is left uniserial.
Proof. By Proposition 3.5 it is sufficient to prove that O/M2 is left uniserial. For
a right uniserial ring O withM2 = 0 (M 6= 0) we have t = 1. By Corollary 3.4M is
a simple left O-module. Therefore, a ring O is uniserial, i.e., right and left uniserial.
As an immediate corollary of this proposition we obtain the following theorem proved
by A. A. Nechaev in [23].
Theorem 3.7 (A. A. Nechaev [23]). A finite ipri-ring is a principal ideal ring (not
only pri-ring).
For more details on finite uniserial (chain) rings see [5].
Acknowledgements. The third author thanks the Department of Mathematics of the
University of São Paulo, Brazil for its warm hospitality during his visit in 2008.
The authors are deeply grateful to Prof. M. Ya. Komarnitskii for his valuable
comments.
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Received 27.11.09
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|
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | Ukrainian English |
| last_indexed | 2026-03-24T02:32:22Z |
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| spelling | umjimathkievua-article-28922020-03-18T19:39:51Z Semiperfect ipri-rings and right Bézout rings Напівдосконалі іпрі-кільця та праві кільця Безу Gubareni, N. M. Dokuchaev, M. A. Kirichenko, V. V. Губарені, Н. М. Докучаєв, М. А. Кириченко, В. В. We present a survey of some results on ipri-rings and right Bézout rings. All these rings are generalizations of principal ideal rings. From the general point of view, decomposition theorems are proved for semiperfect ipri-rings and right Bézout rings. Наведено огляд результатів з теорії іпрі-кілець та правих кілець Безу. Ці кільця є узагальненням кілець головних ідеалів. Із загальної точки зору доведено теореми розкладу для напівдосконалих іпрі-кілець та правих кілець Безу. Institute of Mathematics, NAS of Ukraine 2010-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2892 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 5 (2010); 612–624 Український математичний журнал; Том 62 № 5 (2010); 612–624 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2892/2535 https://umj.imath.kiev.ua/index.php/umj/article/view/2892/2536 Copyright (c) 2010 Gubareni N. M.; Dokuchaev M. A.; Kirichenko V. V. |
| spellingShingle | Gubareni, N. M. Dokuchaev, M. A. Kirichenko, V. V. Губарені, Н. М. Докучаєв, М. А. Кириченко, В. В. Semiperfect ipri-rings and right Bézout rings |
| title | Semiperfect ipri-rings and right Bézout rings |
| title_alt | Напівдосконалі іпрі-кільця та праві кільця Безу |
| title_full | Semiperfect ipri-rings and right Bézout rings |
| title_fullStr | Semiperfect ipri-rings and right Bézout rings |
| title_full_unstemmed | Semiperfect ipri-rings and right Bézout rings |
| title_short | Semiperfect ipri-rings and right Bézout rings |
| title_sort | semiperfect ipri-rings and right bézout rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2892 |
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