C2 Property of Column Finite Matrix Rings
A ring R is called a right C2 ring if any right ideal of R isomorphic to a direct summand of RR is also a direct summand. The ring R is called a right C3 ring if any sum of two independent summands of R is also a direct summand. It is well known that a right C2 ring must be a right C3 ring but the c...
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irk-123456789-1663222020-02-21T15:02:37Z C2 Property of Column Finite Matrix Rings Liang Shen Jianlong Chen Короткі повідомлення A ring R is called a right C2 ring if any right ideal of R isomorphic to a direct summand of RR is also a direct summand. The ring R is called a right C3 ring if any sum of two independent summands of R is also a direct summand. It is well known that a right C2 ring must be a right C3 ring but the converse assertion is not true. The ring R is called J -regular if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. Let N be the set of natural numbers and let Λ be any infinite set. The following assertions are proved to be equivalent for a ring R: (1) CFMFMN(R) is a right C2 ring; (2) CFMFMΛ(R) is a right C2 ring; (3) CFMFMN(R) is a right C3 ring; (4) CFMFMΛ(R) is a right C3 ring; (5) CFMFMN(R) is a J -regular ring and Mn(R) is a right C2 (or right C3) ring for all integers n≥1. Кільце R називається правим C2 кільцем, якщо будь-який правий ідеал R, що є ізоморФним до прямого доданка в RR, також є прямим доданком. Кільце R називається правим C3 кільцем, якщо будь-яка сума двох незалежних доданків в RR також є прямим доданком. Відомо, що праве C2 кільце має бути правим C3 кільцем, але прoтилежне твердження є невірним. Кільце R називається J -регулярним, якщо R/J(R) є регулярним у сенсі фон Ноймана, де J(R) — радикал Якобсона для R. Нехай N — множина натуральних чисел, а Λ — деяка нескінченна множина. Доведено, що наступні твердження є еквівалентними для кільця R: (1) CFMFMN(R) — праве C2 кільце; (2) CFMFMΛ(R) — праве C2 кільце; (3) CFMFMN(R) — праве C3 кільце; (4) CFMFMΛ(R) — праве C3 кільце; (5) CFMFMN(R) — J-регулярне кільце, а Mn(R) — праве C2 (або праве C3) кільце для всіх цілих n > 1. 2014 Article C2 Property of Column Finite Matrix Rings / Liang Shen, Jianlong Chen // Український математичний журнал. — 2014. — Т. 66, № 12. — С. 1718–1722. — Бібліогр.: 6 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166322 512.5 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення |
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Короткі повідомлення Короткі повідомлення Liang Shen Jianlong Chen C2 Property of Column Finite Matrix Rings Український математичний журнал |
| description |
A ring R is called a right C2 ring if any right ideal of R isomorphic to a direct summand of RR is also a direct summand. The ring R is called a right C3 ring if any sum of two independent summands of R is also a direct summand. It is well known that a right C2 ring must be a right C3 ring but the converse assertion is not true. The ring R is called J -regular if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. Let N be the set of natural numbers and let Λ be any infinite set. The following assertions are proved to be equivalent for a ring R:
(1) CFMFMN(R) is a right C2 ring;
(2) CFMFMΛ(R) is a right C2 ring;
(3) CFMFMN(R) is a right C3 ring;
(4) CFMFMΛ(R) is a right C3 ring;
(5) CFMFMN(R) is a J -regular ring and Mn(R) is a right C2 (or right C3) ring for all integers n≥1. |
| format |
Article |
| author |
Liang Shen Jianlong Chen |
| author_facet |
Liang Shen Jianlong Chen |
| author_sort |
Liang Shen |
| title |
C2 Property of Column Finite Matrix Rings |
| title_short |
C2 Property of Column Finite Matrix Rings |
| title_full |
C2 Property of Column Finite Matrix Rings |
| title_fullStr |
C2 Property of Column Finite Matrix Rings |
| title_full_unstemmed |
C2 Property of Column Finite Matrix Rings |
| title_sort |
c2 property of column finite matrix rings |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| topic_facet |
Короткі повідомлення |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166322 |
| citation_txt |
C2 Property of Column Finite Matrix Rings / Liang Shen, Jianlong Chen // Український математичний журнал. — 2014. — Т. 66, № 12. — С. 1718–1722. — Бібліогр.: 6 назв. — англ. |
| series |
Український математичний журнал |
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AT liangshen c2propertyofcolumnfinitematrixrings AT jianlongchen c2propertyofcolumnfinitematrixrings |
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2025-07-14T21:08:58Z |
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2025-07-14T21:08:58Z |
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1837658107652603904 |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
Liang Shen, Jianlong Chen (Southeast Univ., Nanjing, China)
C2 PROPERTY OF COLUMN FINITE MATRIX RINGS*
C2 ВЛАСТИВIСТЬ СТОВПЧИКОВИХ СКIНЧЕННИХ МАТРИЧНИХ КIЛЕЦЬ
A ring R is called a right C2 ring if any right ideal of R isomorphic to a direct summand of RR is also a direct summand.
The ring R is called a right C3 ring if any sum of two independent summands of RR is also a direct summand. It is well
known that a right C2 ring must be a right C3 ring but the converse assertion is not true. The ring R is called J-regular
if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. Let N be the set of natural numbers and Λ
be any infinite set. The following assertions are proved to be equivalent for a ring R : (1) CFMN(R) is a right C2 ring;
(2) CFMΛ(R) is a right C2 ring; (3) CFMN(R) is a right C3 ring; (4) CFMΛ(R) is a right C3 ring; (5) CFMN(R) is a
J-regular ring and Mn(R) is a right C2 (or right C3) ring for all integers n ≥ 1.
Кiльце R називається правим C2 кiльцем, якщо будь-який правий iдеал R, що є iзоморфним до прямого доданка
в RR, також є прямим доданком. Кiльце R називається правим C3 кiльцем, якщо будь-яка сума двох незалежних
доданкiв в RR також є прямим доданком. Вiдомо, що праве C2 кiльце має бути правим C3 кiльцем, але прoтилежне
твердження є невiрним. Кiльце R називається J-регулярним, якщо R/J(R) є регулярним у сенсi фон Ноймана,
де J(R) — радикал Якобсона для R. Нехай N — множина натуральних чисел, а Λ — деяка нескiнченна множина.
Доведено, що наступнi твердження є еквiвалентними для кiльця R: (1) CFMN(R) — праве C2 кiльце; (2) CFMΛ(R)
— праве C2 кiльце; (3) CFMN(R) — праве C3 кiльце; (4) CFMΛ(R) — праве C3 кiльце; (5) CFMN(R) — J-регулярне
кiльце, а Mn(R) — праве C2 (або праве C3) кiльце для всiх цiлих n ≥ 1.
1. Introduction. Throughout this paper, rings are associative with identity and modules are unitary
modules. We denote by N the set of natural numbers. For a ring R, Mn(R) denotes the ring of
all (n × n)-matrices over R and J(R) means the Jacobson radical of R. Let Λ be an infinite set.
CFMΛ(R) means the column finite card(Λ)× card(Λ) matrix ring over a ring R, where card(Λ) is
the cardinality of Λ. For a module M, M (A) is the direct sum of copies of M indexed by a set A.
We use N ≤⊕ M to show that N is a direct summand of M. And use End(M ) to denote the ring of
endomorphisms of M.
The following are three well-known generalizations of the injective condition of a module M.
C1) Every submodule of M is essential in a direct summand of M.
C2) Every submodule that is isomorphic to a direct summand of M is itself a direct summand
of M.
C3) If A and B are direct summands of M with A ∩ B = 0, then A ⊕ B ≤⊕ M. M is called
a Ci module if it satisfies condition Ci, i = 1, 2, 3. C1 modules are also called CS (or extending)
modules. A C2 module is always a C3 module and the converse is not true. A ring R is called a
right Ci ring if the right R-module RR is a Ci module, i = 1, 2, 3. Much more information about
these conditions can be referred to [5].
* The first author is supported by NSF of Jiangsu Province (No.BK20130599) and the Project-sponsored by SRF for
ROCS, SEM. The second author is supported by NSF of China (No. 11371089), NSF of Jiangsu Province (No.20141327),
and Specialized Research Fund for the Doctoral Program of Higher Education (No.20120092110020).
c© LIANG SHEN, JIANLONG CHEN, 2014
1718 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
C2 PROPERTY OF COLUMN FINITE MATRIX RINGS 1719
Let R be a ring and Λ be an infinite set whose cardinality is not ℵ0. It can be proved that
CFMN(R) is a right C1 ring may not inform that CFMΛ(R) is a right C1 ring (see [4], Example). In
this short article, we concentrate on the C2 property of column finite matrix rings. Some interesting
results are obtained. It is proved in Theorem 2.3 that, for any infinite set Λ, CFMN(R) is a right C2
ring if and only if CFMΛ(R) is a right C2 ring if and only if CFMN(R) is a right C3 ring if and
only if CFMΛ(R) is a right C3 ring.
2. Results. First we look at some basic results on column finite matrix rings. Let R be a ring
and Λ be an infinite set. We consider the right R-module R(Λ)
R as the set of all card(Λ)× 1 column
matrices with finite nonzero entries in R. We have the following results.
Proposition 2.1. Let R be a ring and Λ be an infinite set. Then every right ideal I of CFMΛ(R)
has the form I=
{
[α β γ . . .]|α, β, γ, . . . ∈ T
}
, where T is a submodule of R(Λ)
R . In particular, I is
an essential right ideal of CFMΛ(R) if and only if T is an essential submodule of R(Λ)
R , and I is a
direct summand of CFMΛ(R)CFMΛ(R) if and only if T is a direct summand of R(Λ)
R .
Proof. Set A=
{
[α β γ . . .]|α, β, γ, . . . ∈ T
}
, where T is a submodule of R(Λ)
R . It is easy to
verify that A is a right ideal of CFMΛ(R). Now let T be the set of columns those appear in all the
matrices of I. It is clear that T is a submodule of R(Λ)
R and I=
{
[α β γ . . .]|α, β, γ, . . . ∈ T
}
.
Proposition 2.2. Let R be a ring and Λ be an infinite set. Assume e2 = e ∈ R. Set M = eR
and S = eRe. Then End(M (Λ)
R )∼= CFMΛ(S).
Proof. We prove the case Λ = N. The others are similar. To be convenient, we consider
M
(N)
R as the set of all column (N × 1)-matrices with finite nonzero entries in M. Then for any
α ∈ M (N)
R and A ∈ CFMN(S), Aα ∈ M (N)
R . Now define a map F from CFMN(S) to End
(
M
(N)
R
)
such that for every A ∈ CFMN(S) and any α ∈ M (N)
R , F (A)(α) = Aα. It is clear that F is a ring
homomorphism from CFMN(S) to End(M (N)
R ). Next we show that F is an isomorphism. It is easy
to see that F is a monomorphism. We only need to show that F is epic. Let εi be the element
in M (N)
R with the ith entry equal to e and the others are zero, ∀i ∈ N. Assume ϕ ∈ End
(
M
(N)
R
)
.
Let B =
[
ϕ(ε1), ϕ(ε2), . . . , ϕ(εn), . . .
]
and E =
[
ε1, ε2, . . . , εn, . . .
]
be the matrices with the ith
column equal to ϕ(εi) and εi, respectively, i ∈ N. It is clear that E2 = E and BE ∈ CFMN(S).
For each X ∈M (N)
R , there exists finite nonzero elements ri ∈ eR, i ∈ N, such that X =
∑∞
i=1
εiri.
Let C be the column (N × 1)-matrix with the ith entry equal to ri, i ∈ N. Then X = EC. Thus
ϕ(X) = ϕ
(∑∞
i=1
εiri
)
=
∑∞
i=1
ϕ(εi)ri =
∑∞
i=1
ϕ(εi)eri = BEC = BEEC = BEX. Set
A = BE. It is clear that ϕ = F (A). Therefore, F is an epimorphism.
Lemma 2.1 ([6], Theorem 7.14). Let MR be a module and write E = End(MR). Then
(1) If E is a right C2 ring, then MR is a C2 module.
(2) The converse in (1) holds if Ker(α) is generated by M whenever α ∈ E is such that rE(α)
is a direct summand of EE .
Theorem 2.1. Let R be a ring and Λ be an infinite set. Then
(1) CFMΛ(R) is right C1 if and only if R(Λ)
R is a C1 module.
(2) CFMΛ(R) is right C2 if and only if R(Λ)
R is a C2 module.
(3) CFMΛ(R) is right C3 if and only if R(Λ)
R is a C3 module.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1720 LIANG SHEN, JIANLONG CHEN
Proof. (1) and (3) are directly obtained by Proposition 2.1.
(2) By Proposition 2.2, End(R(Λ)
R ) ∼= CFMΛ(R). Since R(Λ)
R is a generator of right R-modules,
according to the above lemma, CFMΛ(R) is a right C2 ring if and only if R(Λ)
R is a C2 module.
Applying a similar proof, we have the following theorem.
Theorem 2.2. Let R be a ring and n be a positive integer. Consider RnR as direct sum of n
copies of RR. Then
(1) Mn(R) is right C1 if and only if RnR is a C1 module.
(2) Mn(R) is right C2 if and only if RnR is a C2 module.
(3) Mn(R) is right C3 if and only if RnR is a C3 module.
Recall that a ring R is called right (countably) Σ-CS if every (countable) direct sum of copies of
RR is CS. And a right countably Σ-CS ring may not be right Σ-CS. In fact, a von Neumann regular
right self-injective ring is right countably Σ-CS but not right Σ-CS unless it is semisimple (see [4],
Example). Thus, by Theorem 2.1, CFMN(R) is a right C1 ring may not imply that CFMΛ(R) is a
right C1 ring for every infinite set Λ. But if C1 is replaced by C2 or C3, the results will be different
and interesting. Before giving our main results, we need some lemmas.
The next result was firstly obtained by Yiqiang Zhou. To be self-contained, we write down the
proof.
Lemma 2.2 (Zhou’s lemma). Let R be a ring and M be a right R-module. If the direct sum
M ⊕M is a C3 module, then M is a C2 module.
Proof. Assume K is a submodule of M that is isomorphic to a direct summand L of M. We
want to show that K is also a direct summand of M. Let f be the isomorphism from K to L. Set
K ′ = {(x, f(x)) : x ∈ K}, L′ = 0 ⊕ L and M ′ = M ⊕ 0. Then K ′ + M ′ = M ⊕ L is a direct
summand of M ⊕M. Since K ′ ∩M ′ = 0, K ′ is also a direct summand of M ⊕M. It is clear that
K ′∩L′ = 0 and L′ is a direct summand ofM⊕M. BecauseM⊕M is a C3 module, K ′+L′ = K⊕L
is a direct summand of M ⊕M. As K ⊕ 0 is a direct summand of K ⊕ L, K ⊕ 0 is also a direct
summand of M ⊕M. This shows that K ⊕ 0 is a direct summand of M ⊕ 0. It is clear that K is a
direct summand of M.
We define a ring R to be J-regular if R/J(R) is a von Neumann regular ring.
Lemma 2.3. A ring R is right perfect if and only if CFMN(R) is a J-regular ring.
Proof. See [3], Theorem 1.
Lemma 2.4 ([1], Lemma 19.18). Let R be a ring and V be a flat right R-module and suppose
that the sequence
0→ K → V → V
′ → 0
is exact. Then V
′
is flat if and only if for each (finitely generated) left ideal I ⊆RR, KI = K ∩ V I.
Theorem 2.3. The following are equivalent for a ring R.
(1) CFMN(R) is a right C2 ring.
(2) CFMN(R) is a right C3 ring.
(3) For any infinite set Λ, CFMΛ(R) is a right C2 ring.
(4) For any infinite set Λ, CFMΛ(R) is a right C3 ring.
(5) CFMN(R) is a J-regular ring and Mn(R) is right C2 for all integer n ≥ 1.
(6) CFMN(R) is a J-regular ring and Mn(R) is right C3 for all integer n ≥ 1.
Proof. Let Λ be an infinite set. It is clear that R(Λ)
R
∼= (R
(Λ)
R ⊕ R(Λ)
R ). Then by Theorem 2.1,
Theorem 2.2 and Lemma 2.2, we have (1) ⇔ (2), (3) ⇔ (4) and (5) ⇔ (6). Next we only need to
prove (1) ⇒ (5)⇒ (3) ⇒ (1).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
C2 PROPERTY OF COLUMN FINITE MATRIX RINGS 1721
(1)⇒ (5). If R satisfies (1), by Theorem 2.1, R(N)
R is a C2 module. For any integer n ≥ 1, RnR
can be looked on as a direct summand of R(N)
R . Since a direct summand of a C2 module is always
a C2 module, we have that RnR is a C2 module. Then by Theorem 2.2, Mn(R) is right C2 for all
integer n ≥ 1. Now we prove that CFMN(R) is a J-regular ring. According to Lemma 2.3, we need
to show that R is a right perfect ring. By [1] (Theorem 28.4), we will prove that R satisfies DCC
on principal left ideals of R. The following method is owing to Bass [2]. Let Ra1 ⊇ Ra2a1 ⊇ . . .
be any descending chain of principal left ideals of R. Set F = R
(N)
R with free basis x1, x2, . . . and G
be the submodule of F spanned by yi = xi − xi+1ai, i ∈ N. By [1] (Lemma 28.1), G is free with
basis y1, y2, . . . . Thus G ∼= F. F is a C2 module implies that G is a direct summand of F. Then by
[1] (Lemma 28.2), the chain Ra1 ⊇ Ra2a1 ⊇ . . . terminates.
(5) ⇒ (3). By Theorem 2.1, We only need to show that R(Λ)
R is a C2 module. Assume K
is a submodule of the free module F = R
(Λ)
R and K is isomorphic to a direct summand of F. In
order to show that K is also a direct summand of F, we only need to prove that F/K is a projective
R-module. Since CFMN(R) is a J-regular ring, by Lemma 2.3, R is a right perfect ring. According
to [1] (Theorem 28.4), every flat right R-module is projective. Thus, we just need to show that F/K
is flat. As R is right perfect, R is semiperfect. Then R has a basic set of primitive idempotents
e1, . . . , em. Since K is projective, by [1] (Theorem 27.11), there exist sets A1, . . . , Am such that
K ∼= (e1R)(A1) ⊕ . . . ⊕ (emR)(Am). Set λ = card(Λ). Since K is isomorphic to a direct summand
of F, K is λ-generated. So each (eiR)(Ai) is also λ-generated, i = 1, 2, . . . ,m. As λ is an infinite
cardinality, by [1] (Lemma 25.7), card(Ai) ≤ λ, i = 1, 2, . . . ,m. So card(A1) + . . .+ card(Am) ≤
≤ mλ = λ. Set L = (e1R)(A1)⊕ . . .⊕(emR)(Am). Then L can be considered as a direct summand of
F. Let A =
{
Lα ≤⊕ L : Lα ∼= (e1R)(Aα1 ) ⊕ . . .⊕ (emR)(Aαm ) with card(Aα1) + . . .+ card(Aαm)
is finite
}
. It is clear that L =
⋃
Lα∈A Lα and, for any left ideal I of R, LI =
⋃
Lα∈A LαI. Now let
f be the isomorphism from K to L. Set B =
{
Kα = f−1(Lα) : Lα ∈ A
}
. Since K is isomorphic to
L, K =
⋃
Kα∈BKα and, for any left ideal I of R, KI =
⋃
Kα∈BKαI. By Theorem 2.2, RnR is a C2
module for all integers n ≥ 1. As Lα is a finitely generated direct summand of L for each Lα ∈ A,
it is easy to verify that Kα is a direct summand of F for each Kα ∈ B. At last we apply Lemma 2.4
to show that F/K is a flat module. Let I be any left ideal of R, by Lemma 2.4, Kα ∩ FI = KαI,
Kα ∈ B. Then K ∩ FI = (
⋃
Kα∈BKα) ∩ FI =
⋃
Kα∈B(Kα ∩ FI) =
⋃
Kα∈BKαI = KI. Thus,
by Lemma 2.4, F/K is flat.
(3) ⇒ (1). If R satisfies (3), by Theorem 2.1, R(Λ)
R is a C2 module. Since Λ is an infinite set,
R
(N)
R can be looked on as a direct summand of R(Λ)
R . As a direct summand of C2 module is always
C2, we have R(N)
R is a C2 module. Applying Theorem 2.1 again, CFMN(R) is a right C2 ring.
Based on Theorem 2.1, Theorem 2.2, Lemma 2.3 and Theorem 2.3, we have the following
corollary.
Corollary 2.1. The following are equivalent for a ring R.
(1) R(N)
R is a C2 module.
(2) R(N)
R is a C3 module.
(3) For any infinite set Λ, R
(Λ)
R is a C2 module.
(4) For any infinite set Λ, R
(Λ)
R is a C3 module.
(5) R is a right perfect ring and every finite direct sum of copies of RR is a C2 module.
(6) R is a right perfect ring and every finite direct sum of copies of RR is a C3 module.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1722 LIANG SHEN, JIANLONG CHEN
Acknowledgements. The article was written during the first author’s visiting Center of Ring
Theory and Its Applications in Department of Mathematics, Ohio University. He would like to thank
the center for the hospitality. The authors are grateful to Professor Dinh Van Huynh, Professor Sergio
R. López-Permouth, Professor Yiqiang Zhou and Dr Gangyong Lee for their helpful suggestions.
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Received 08.11.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
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