On one class of semiperfect semidistributive rings
In this paper we consider the Artinian semidistributive rings.
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Інститут прикладної математики і механіки НАН України
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| Cite this: | On one class of semiperfect semidistributive rings / M. Kasyanuk // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 19–22. — Бібліогр.: 2 назв. — англ. |
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Kasyanuk, M. 2019-06-09T13:41:31Z 2019-06-09T13:41:31Z 2013 On one class of semiperfect semidistributive rings / M. Kasyanuk // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 19–22. — Бібліогр.: 2 назв. — англ. 1726-3255 2010 MSC:16P40, 16G10. https://nasplib.isofts.kiev.ua/handle/123456789/152260 In this paper we consider the Artinian semidistributive rings. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On one class of semiperfect semidistributive rings Article published earlier |
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Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України |
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In this paper we consider the Artinian semidistributive rings.
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On one class of semiperfect semidistributive rings / M. Kasyanuk // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 19–22. — Бібліогр.: 2 назв. — англ. |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 15 (2013). Number 1. pp. 19 – 22
c© Journal “Algebra and Discrete Mathematics”
On one class
of semiperfect semidistributive rings
Marina Kasyanuk
Communicated by A. V. Zhuchok
Abstract. In this paper we consider the Artinian semidis-
tributive rings.
1. Introduction
It is well known that many important classes of rings are naturally
characterized by the properties of modules over them. As examples, we
mention semiperfect semidistributive rings.The first papers on the theory
of semidistributive rings were appeared in the middle of XX century.
The reduction theorem for SPSD-rings and decomposition theorem for
semiprime right Noetherian SPSD-rings were proved in [2].
2. Reduction theorem for SPSD-rings
We write SPSDR-ring (SPSDL-ring) for a semiperfect right (left)
semidistributive ring and SPSD-ring for a semiperfect semidistributive
ring.
Theorem 1. (A.Tuganbaev). A semiperfect ring A is right (left) semi-
distributive if and only if for any local idempotents e and f of the ring A
the set eAf is a uniserial right fAf-module (uniserial left eAe-module).
2010 MSC: 16P40, 16G10.
Key words and phrases: Q-symmetric ring; semiperfect ring; semidistributive
module; quiver of semiperfect ring.
20 On one class of semiperfect semidistributive rings
Corollary 1. 1 = e1 + ... + en be a decomposition of 1 ∈ A into a
sum of mutually orthogonal local idempotents. The ring A is right (left)
semidistributive if and only if for any idempotents ei and ej of the above
decomposition, the set eiAej is a uniserial right ejAej-module (left eiAei-
module).
Corollary 2. Let A be a semiperfect ring, and let 1 = e1 + ... + en
be a decomposition of 1 ∈ A into a sum of mutually orthogonal local
idempotents. The ring A is right (left) semidistributive if and only if for
any idempotents ei and ej (i = j) of the above decomposition the ring
(ei + ej)A(ei + ej) is right (left) semidistributive.
Corollary 3. Let A be a Noetherian SPSD-ring, and let 1 = e1 + ...+en
be a decomposition of the identity 1 ∈ A into a sum of mutually orthogonal
local idempotents, let Aij = eiAej and let Ri be the Jacobson radical of a
ring Aii. Then RiAij = AijRj for i, j = 1, ..., n.
Example 1. Consider A =
(
R C
0 C
)
as an R-algebra (R is the field
of real numbers, C is the field of complex numbers). The Peirce decom-
position of the Jacobson radical R = R(A) has the form R =
(
0 C
0 0
)
and the R-algebra A is right serial, i.e., right semidistributive. The left
indecomposable projective Q2 =
(
C
C
)
has socle
(
C
0
)
, which is a
direct sum of two copies of the left simple module
(
R
0
)
. Consequently,
the R-algebra A is an SPSDR-ring but it is not an SPSDL-ring.
3. Quivers of SPSD-rings
Recall that a quiver without multiple arrows and multiple loops is
called a simply laced quiver. Let A be an SPSD-ring. The quotient ring
A/R2 is right Artinian and its quiver Q(A) is defined by Q(A) = Q(A/R2).
Theorem 2. The quiver Q(A) of an SPSD-ring A is simply laced. Con-
versely, for any simply laced quiver Q there exists an SPSD-ring A such
that Q(A) = Q.
Corollary 4. The link graph LG(A) of an SPSD-ring A coincides with
a Q(A).
M. Kasyanuk 21
Proof. For any SPSD-ring A the following equalities hold: LG(A) =
Q(A, R) = Q(A).
Theorem 3. For an Artinian ring A with R2 = 0 the following conditions
are equivalent: (a) A is semidistributive; (b) Q(A) is simply laced and the
left quiver Q(A) can be obtained from Q(A) by reversing all arrows.
Definition 1. A semiperfect ring A such that A/R2 is Artinian will be
called Q-symmetric if the left quiver Q(A) can be obtained from the right
quiver Q(A) by reversing all arrows.
Corollary 5. Every SPSD-ring is Q-symmetric.
Note 1. Example 1 shows that an SPSDR-ring is not always Q- sym-
metric.
Let O be a discrete valuation ring with an unique maximal ideal
M = πO = Oπ Then all ideals O( left, right, two- sided) limited powers
Mk = πkO = Oπk.
Denote by Mn(Z) the ring of all square n×n-matrices over the ring of
integers Z. Let E = Mn(Z). We shall call a matrix E = (aij) an exponent
matrix if αij + αjk ≥ αik for i, j, k = 1, ..., n and aii = 0 for i = 1, ..., n.
A matrix E is called a reduced exponent matrix if αij + αji > 0 for
i, j = 1, ..., n. We shall use the following notation: A = {O, E(A)}, where
E(A) = (αij) is the exponent matrix of a ring A, i.e.,
A =
n
∑
i,j=1
eijπαijO,
, where the eij are the matrix units. If a tiled order is reduced, then
αij + αji > 0 for i, j = 1, ..., n, i 6= j, i.e., E(A) is reduced.
Theorem 4. The ring A = {O, E(A)} is Artinian semidistributive ring.
Theorem 5. Let A be a semiperfect semidistributive ring and AA =
P n1
1
⊕ . . . ⊕ P ns
s be a decomposition of a regular module AA into a direct
sum of indecomposable pojective A-modules. A ring A is Artinian if and
only if all endomorphism rings of Pi, (i = 1, . . . , n) are Artininan.
Proof follows from theorem 1.
References
[1] Michiel Hazewinkel, Nadya Gubareni, V. V. Kirichenko Algebras, Rings and
Modules, volume 1, Netherlands: Springer.-2007.- p. 343-353
[2] V.V. Kirichenko and M.A.Khibina Semi-perfect semi-distributive rings, In: Insti-
tute Groups and Related Algebraik Topics, Institute of Mathematics NAS Ukraine,
1993, pp. 457-480
22 On one class of semiperfect semidistributive rings
Contact information
M. Kasyanuk Department of Mechanics and Mathemat-
ics, Kyiv National Taras Shevchenko Univ.,
Volodymyrska str., 64, 01033 Kyiv, Ukraine
E-Mail: marinasayenko@gmail.com
URL: http://www.mechmat.univ.kiev.ua
Received by the editors: 17.02.2013
and in final form 17.02.2013.
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