Modules in which every surjective endomorphism has a δ-small kernel
In this paper,we introduce the notion of δ-Hopfian modules. We give some properties of these modules and provide a characterization of semisimple rings in terms of δ-Hopfian modules by proving that a ring R is semisimple if and only if every R-module is δ-Hopfian. Also, we show that for a ring R, δ(...
Збережено в:
| Дата: | 2018 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2018
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/188409 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Modules in which every surjective endomorphism has a δ-small kernel / S. Ebrahimi Atani, M. Khoramdel, S. Dolati Pishhesari // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 170–189. — Бібліогр.: 18 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-188409 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1884092023-02-28T01:27:02Z Modules in which every surjective endomorphism has a δ-small kernel Ebrahimi Atani, S. Khoramdel, M. Dolati Pishhesari, S. In this paper,we introduce the notion of δ-Hopfian modules. We give some properties of these modules and provide a characterization of semisimple rings in terms of δ-Hopfian modules by proving that a ring R is semisimple if and only if every R-module is δ-Hopfian. Also, we show that for a ring R, δ(R) = J(R) if and only if for all R-modules, the conditions δ-Hopfian and generalized Hopfian are equivalent. Moreover, we prove that δ-Hopfian property is a Morita invariant. Further, the δ-Hopficity of modules over truncated polynomial and triangular matrix rings are considered. 2018 Article Modules in which every surjective endomorphism has a δ-small kernel / S. Ebrahimi Atani, M. Khoramdel, S. Dolati Pishhesari // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 170–189. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC: 16D10, 16D40, 16D90. http://dspace.nbuv.gov.ua/handle/123456789/188409 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper,we introduce the notion of δ-Hopfian modules. We give some properties of these modules and provide a characterization of semisimple rings in terms of δ-Hopfian modules by proving that a ring R is semisimple if and only if every R-module is δ-Hopfian. Also, we show that for a ring R, δ(R) = J(R) if and only if for all R-modules, the conditions δ-Hopfian and generalized Hopfian are equivalent. Moreover, we prove that δ-Hopfian property is a Morita invariant. Further, the δ-Hopficity of modules over truncated polynomial and triangular matrix rings are considered. |
| format |
Article |
| author |
Ebrahimi Atani, S. Khoramdel, M. Dolati Pishhesari, S. |
| spellingShingle |
Ebrahimi Atani, S. Khoramdel, M. Dolati Pishhesari, S. Modules in which every surjective endomorphism has a δ-small kernel Algebra and Discrete Mathematics |
| author_facet |
Ebrahimi Atani, S. Khoramdel, M. Dolati Pishhesari, S. |
| author_sort |
Ebrahimi Atani, S. |
| title |
Modules in which every surjective endomorphism has a δ-small kernel |
| title_short |
Modules in which every surjective endomorphism has a δ-small kernel |
| title_full |
Modules in which every surjective endomorphism has a δ-small kernel |
| title_fullStr |
Modules in which every surjective endomorphism has a δ-small kernel |
| title_full_unstemmed |
Modules in which every surjective endomorphism has a δ-small kernel |
| title_sort |
modules in which every surjective endomorphism has a δ-small kernel |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/188409 |
| citation_txt |
Modules in which every surjective endomorphism has a δ-small kernel / S. Ebrahimi Atani, M. Khoramdel, S. Dolati Pishhesari // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 2. — С. 170–189. — Бібліогр.: 18 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT ebrahimiatanis modulesinwhicheverysurjectiveendomorphismhasadsmallkernel AT khoramdelm modulesinwhicheverysurjectiveendomorphismhasadsmallkernel AT dolatipishhesaris modulesinwhicheverysurjectiveendomorphismhasadsmallkernel |
| first_indexed |
2025-07-16T10:26:28Z |
| last_indexed |
2025-07-16T10:26:28Z |
| _version_ |
1837798879443025920 |
| fulltext |
“adm-n4” — 2019/1/24 — 10:02 — page 170 — #20
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 26 (2018). Number 2, pp. 170–189
c© Journal “Algebra and Discrete Mathematics”
Modules in which every surjective endomorphism
has a δ-small kernel
Shahabaddin Ebrahimi Atani, Mehdi Khoramdel
and Saboura Dolati Pishhesari
Communicated by R. Wisbauer
Abstract. In this paper, we introduce the notion of δ-Hopfian
modules. We give some properties of these modules and provide
a characterization of semisimple rings in terms of δ-Hopfian modules
by proving that a ring R is semisimple if and only if every R-module
is δ-Hopfian. Also, we show that for a ring R, δ(R) = J(R) if and
only if for all R-modules, the conditions δ-Hopfian and generalized
Hopfian are equivalent. Moreover, we prove that δ-Hopfian prop-
erty is a Morita invariant. Further, the δ-Hopficity of modules over
truncated polynomial and triangular matrix rings are considered.
Introduction
Throughout rings will have unity and modules will be unitary. Let M
denote a right module over a ring R. The study of modules by properties
of their endomorphisms has long been of interest. The concept of Hopfian
groups was introduced by Baumslag in 1963 ([2]). In [8], Hiremath general-
ized this concept to general module theoretic setting. A right R-module M
is called Hopfian, if any surjective endomorphism of M is an isomorphism.
Later, the dual concept of Hopfian modules (co-Hopfian modules) was
introduced. Hopfian and co-Hopfian modules (rings) have been investi-
gated by several authors [4], [7], [14], [15] and [17]. Direct finiteness (or
2010 MSC: 16D10, 16D40, 16D90.
Key words and phrases: Dedekind finite modules, Hopfian modules, generalized
Hopfian modules, δ-Hopfian modules.
“adm-n4” — 2019/1/24 — 10:02 — page 171 — #21
S. E. Atani, M. Khoramdel, S. D. Pishhesari 171
Dedekind finiteness) evolved from the concepts of “finite projections” in
operator algebras and “finite idempotents” in Baer rings (see [5, p. 74]
and [9, p.10]). A module M is called Dedekind-finite, if X = 0 is the only
module for which M ∼= M ⊕X. Equivalently, fg = 1 implies gf = 1 for
each f, g ∈ End(M).
In [4], a proper generalization of Hopfian modules, called generalized
Hopfian modules, was given. A right R-module M is called generalized
Hopfian, if any surjective endomorphism of M has a small kernel. Recall
that a submodule N of a module M is called small, denoted by N ≪ M , if
N +X 6= M for all proper submodules X of M . In [4, Corollary 1.4], it is
shown that the concepts of Dedekind finite modules, Hopfian modules and
generalized Hopfian modules coincide for every (quasi-)projective module.
A submodule N of a module M is called δ-small in M, written N ≪δ M ,
provided N + K 6= M for any proper submodule K of M with M/K
singular (see [18]). In this paper, we introduce and study the notion of
δ- Hopfian modules, which is a generalization of Hopfian modules and
generalized Hopfian modules. We replace “small kernel” by “δ-small kernel”.
We discuss the following questions: When does a module have the property
that every of its surjective endomorphisms has a δ-small kernel? Further,
how can δ-Hopfian modules be used to characterize the base ring itself?
We summarize the contents of this article as follows. In Section 2,
we give some equivalent properties and characterizations of δ-Hopfian
modules. We characterize semisimple rings in terms of δ-Hopfian modules
and show that a ring R is semisimple if and only if every R-module
is δ-Hopfian. We prove that for a ring R, δ(R) = J(R) if and only if
for all R-modules, the conditions δ-Hopfian and generalized Hopfian are
equivalent. It is shown that a direct sum of δ-Hopfian modules and their
endomorphism rings need not have the same property. Also, we prove that
δ-Hopfian property is a Morita invariant.
In Section 3, we consider the δ-Hopfian property of M [x] (as an
R[x]-module) and M [x]/(xn+1) (as an R[x]/(xn+1)-module). We char-
acterize the structures of maximal submodules, essential submodules of
M [x]/(xn+1). Also, we show that
δ(M [x]/(xn+1)) = Rad(M) +Mx+Mx2 + · · ·+Mxn.
Moreover, we prove that if M [x]/(xn+1) is δ-Hopfian as an R[x]/(xn+1)-
module, then M is δ-Hopfian, but the converse is not true.
In Section 3, we characterize the generalized triangular matrix rings
which are right δ-Hopfian and prove that if M is an (S,R)-bimodule, and
“adm-n4” — 2019/1/24 — 10:02 — page 172 — #22
172 δ-Hopfian modules
T =
(
S M
0 R
)
, then δ(TT ) =
(
H M
0 δ(RR)
)
, where,
H = δ(SS) ∩ {I : I is a maximal right ideal of S with annS(M) ⊆ I}.
At the end of the paper, some open problems are given.
We now fix our notations and state a few well known preliminary results
that will be needed. Let M be a right R-module. For submodules N and K
of M , N 6 K denotes that N is a submodule of K, Rad(M) denotes the
Jacobson radical of M and End(M) denotes the ring of endomorphisms
of M . By N 6ess M , we mean that N is an essential submodule of M .
Also, for a module M and a set Λ, let M (Λ) denote the direct sum of
|Λ| copies of M , where |Λ| is the cardinality of Λ. The symbols J(R),
Mn(R) and Tn(R) denote the Jacobson radical of R, the full ring of n-by-n
matrices over R, and the ring of n-by-n upper triangular matrices over R,
respectively. As in [18], we define δ(M) to be
RejP(M) = ∩{N 6 M : M/N ∈ P},
where P is the class of all singular simple modules.
Recall that a ring R is said to satisfy the rank condition if a right
R-epimorphism Rm → Rn can exist only when m > n (see [10]).
Lemma 1 ([18, Lemma 1.2]). Let N be a submodule of M . The following
are equivalent:
(1) N ≪δ M .
(2) If X + N = M , then M = X ⊕ Y for a projective semisimple
submodule Y with Y ⊆ N ;
(3) If X +N = M with M/X Goldie torsion, then X = M .
Lemma 2 ([18, Lemma 1.3]). Let M be a module.
(1) For submodules N,K,L of M with K ⊆ N , we have
(a) N ≪δ M if and only if K ≪δ M and N/K ≪δ M/K.
(b) N + L ≪δ M if and only if N ≪δ M and L ≪δ M .
(2) If K ≪δ M and f : M → N is a homomorphism, then f(K) ≪δ N .
(3) Let K1 6 M1 6 M , K2 6 M2 6 M , and M = M1 ⊕ M2. Then
K1 ⊕K2 ≪δ M1 ⊕M2 if and only if K1 ≪δ M1 and K2 ≪δ M2.
Lemma 3 ([18, Lemma 1.5, Theorem 1.6]). Let R be a ring and M an
R-module. Then δ(M) =
∑
{L 6 M : L ≪δ M} and δ(R) equals the
intersection of all essential maximal right ideals of R.
“adm-n4” — 2019/1/24 — 10:02 — page 173 — #23
S. E. Atani, M. Khoramdel, S. D. Pishhesari 173
Lemma 4 ([3, 5, 10]). Let T =
(
S M
0 R
)
, where R and S are rings and
M is an (S,R)-bimodule.
(1) Every right ideal of T has the form
(
I N
0 J
)
such that I 6 SS,
J 6 RR and IM ⊆ N .
(2) Let Q =
(
I N
0 J
)
. Then Q is a maximal right ideal of T if and only
if N = M and either I = S and J is a maximal right ideal of R or
J = R and I is a maximal right ideal of S.
(3) The right ideal Q =
(
I N
0 J
)
of T is essential in TT if and only if
N 6ess MR, J 6ess RR and I ∩ (annS(M)) 6ess annS(M).
1. δ-Hopfian modules
Motivated by the definition of generalized Hopfian modules, we intro-
duce the key definition of this paper.
Definition 1. Let M be an R-module. We say that M is δ-Hopfian (δH
for short) if any surjective R-endomorphism of M has a δ-small kernel in
M .
The next result gives several equivalent conditions for a δH module.
Theorem 1. Let M be an R-module. The following statements are equiv-
alent:
(1) M is δH;
(2) For any epimorphism f : M → M , if N ≪δ M , then f−1(N) ≪δ M ;
(3) If N 6 M and there is an R-epimorphism M/N → M , then N ≪δ
M ;
(4) If M/N is nonzero and singular for some N 6 M , then f(N) 6= M ,
for each R-surjective endomorphism f of M ;
(5) There exists a fully invariant δ-small submodule N of M such that
M/N is δH;
(6) If f : M → M ⊕X is an epimorphism, where X is a module, then
X is projective and semisimple.
Proof. (1) ⇒ (2) Assume that f : M → M is an epimorphism and N ≪δ
M . Let f−1(N) + K = M for some K 6 M , where M/K is singular.
Hence N + f(K) = M . As M/K is singular and M/f(K) is an image of
M/K, M/f(K) is singular. Hence N + f(K) = M and N ≪δ M , giving
“adm-n4” — 2019/1/24 — 10:02 — page 174 — #24
174 δ-Hopfian modules
f(K) = M . So K +Ker(f) = M . Since M is δH, Ker(f) ≪δ M . Hence
M/K is singular implies that K = M . Thus f−1(N) ≪δ M .
(2) ⇒ (3) Let f : M/N → M be an epimorphism. It is clear that
N 6 (fπ)−1(0), where π : M → M/N is the canonical epimorphism. By
(2), (fπ)−1(0) ≪δ M , Hence by Lemma 2, N ≪δ M .
(3) ⇒ (4) Let N be a proper submodule of M such that M/N is
singular and f a surjective endomorphism of M with f(N) = M . Then
N + Ker(f) = M . Hence Ker(f) ≪δ M by (3), and so N = M , a
contradiction.
(4) ⇒ (1) Let f : M → M be an epimorphism. If M = N + Ker(f),
with M/N is singular, then M = f(M) = f(N). Hence N = M by (4).
Thus Ker(f) ≪δ M .
(1) ⇒ (5) Take N = 0.
(5) ⇒ (1) Let M/N be δH for some fully invariant δ-small submodule
N of M . If f : M → M is an epimorphism, then f̄ : M/N → M/N
with f̄(m + N) = f(m) + N (m ∈ M) is an epimorphism. As M/N
is δH, ker(f̄) ≪δ M/N . Since (Ker(f) + N)/N ⊆ ker(f̄) ≪δ M/N ,
Ker(f) +N ≪δ M by Lemma 2. Hence Ker(f) ≪δ M by Lemma 2, and
so M is δH.
(1) ⇒ (6) Let f : M → M⊕X be an epimorphism, π : M⊕X → M the
natural projection. It is clear that Ker(πf) = f−1(0⊕X). By (1), M is δH.
Hence Ker(πf) ≪δ M . Since f is an epimorphism, f [f−1(0⊕X)] = 0⊕X.
Hence by Lemma 2, 0⊕X = f(Ker(πf)) ≪δ M ⊕X. Therefore X ≪δ X
by Lemma 2. So, by Lemma 1, X is projective and semisimple.
(6) ⇒ (1) Let f be a surjective endomorphism of M and Ker(f)+L =
M for some L 6 M , where M/L is singular. Since
M/Ker(f) ∩ L = Ker(f)/(Ker(f) ∩ L)⊕ L/(Ker(f) ∩ L)
∼= M/L⊕M/Ker(f) ∼= M/L⊕M,
the epimorphism M → M ⊕M/L exists. By (6), M/L is semisimple and
projective. As M/L is singular,M/L = 0. Thus M = L and Ker(f) ≪δ M .
Corollary 1. Let M be a δH module, f ∈ End(M) an epimorphism
and N 6 M . Then N ≪δ M if and only if f(N) ≪δ M if and only if
f−1(N) ≪δ M . Moreover δ(M) =
∑
N≪δM
f(N) =
∑
N≪δM
f−1(N).
Proposition 1. Let M be a δH module. If N is a direct summand of M ,
then N is δH.
“adm-n4” — 2019/1/24 — 10:02 — page 175 — #25
S. E. Atani, M. Khoramdel, S. D. Pishhesari 175
Definition 2. Let M and N be two R-modules. M is called δ-Hopfian (δH,
for short) relative to N , if for each epimorphism f : M → N ,Ker(f) ≪δ M .
In view of the above definition, an R-module M is δH if and only if
M is δH relative to M .
In the following, we characterize the δ-Hopfian modules in terms of
their direct summands and factor modules.
Proposition 2. Let M and N be two R-modules. Then the following
statements are equivalent:
(1) M is δH relative to N ;
(2) for each L 6⊕ M , L is δH relative to N ;
(3) for each L 6 M , M/L is δH relative to N .
Proof. (1) ⇒ (2) Let L 6⊕ M say M = L ⊕ K, where K 6 M and
f : L → M an epimorphism. Let π : M → L be the natural projection.
Then fπ : M → N is an epimorphism and so Ker(fπ) ≪δ M by (1). It is
clear that Ker(fπ) = Ker(f) ⊕K. Thus Ker(fπ) = Ker(f) ⊕K ≪δ M .
By Lemma 2(3), Ker(f) ≪δ L.
(2) ⇒ (1) Take L = M .
(1) ⇒ (3) Let L 6 M and f : M/L → N be an epimorphism. Then
fπ : M → N is an epimorphism, where π : M → M/L is the natu-
ral homomorphism. As Ker(fπ) = π−1(Ker(f)) and Ker(fπ) ≪δ M ,
π(Ker(fπ)) = Ker(f) ≪δ M/L by Lemma 2. Therefore M/L is δH
relative to N .
(3) ⇒ (1) Take L = 0.
In the following, we present some characterizations of projective δH
modules.
Theorem 2. Let M be a projective R-module. Then the following state-
ments are equivalent:
(1) M is δH;
(2) If f ∈ End(M) has a right inverse, then Ker(f) is semisimple and
projective;
(3) If f ∈ End(M) has a right inverse in End(M), then Ker(f) ≪δ M ;
(4) If f ∈ End(M) has a right inverse g, then (1− gf)M ≪δ M .
Proof. Let f ∈ End(M) be an epimorphism. Then there exists g ∈
End(M) such that fg = 1 ∈ End(M). It is clear that Ker(f) = (1−gf)M
and M = Ker(f)⊕ (gf)M .
“adm-n4” — 2019/1/24 — 10:02 — page 176 — #26
176 δ-Hopfian modules
(1) ⇒ (2) Let f ∈ End(M) have a right inverse in End(M). Then fg =
1 for some g ∈ End(M). Thus f is an epimorphism and so Ker(f) ≪δ M .
As Ker(f) = (1− gf)M is a direct summand of M , it is semisimple and
projective by Lemma 1.
(2) ⇒ (3) Let f ∈ End(M) have a right inverse in End(M). Then by
(2), Ker(f) is semisimple and projective. We show that Ker(f) ≪δ M .
Let Ker(f) + L = M for some L 6 M . Since Ker(f) is semisimple,
(Ker(f) ∩ L)⊕ T = Ker(f) for some T 6 Ker(f). Therefore T ⊕ L = M .
As T is semisimple and projective, Ker(f) ≪δ M , by Lemma 1.
(3) ⇒ (4) Let f ∈ End(M) have a right inverse g. Hence by (3),
Ker(f) = (1− gf)M ≪δ M .
(4) ⇒ (1) Let f ∈ End(M) be an epimorphism. As M is projective,
f ∈ End(M) has a right inverse g and Ker(f) = (1− gf)M . Therefore by
(4), Ker(f) ≪δ M and M is δH.
Next, we characterize the class of rings R for which every (free) R-
module is δH.
Theorem 3. Let R be a ring. Then the following statements are equivalent:
(1) Every R-module is δH;
(2) Every projective R-module is δH;
(3) Every free R-module is δH;
(4) R is semisimple.
Proof. (1) ⇒ (2) ⇒ (3) They are clear.
(3) ⇒ (4) By (3), R(N) is δH. As R(N) ∼= R(N) ⊕R(N), by Theorem 1,
R(N) is semisimple. Hence R is semisimple.
(4) ⇒ (1) Let M be an R-module. Hence M is projective and for each
surjective endomorphism f of M , Ker(f) is semisimple and projective.
Hence by Theorem 2, M is δH.
It is clear that every generalized Hopfian module is δH. The following
example shows that the converse is not true, in general. Also, it shows
that a δH module need not be Dedekind-finite.
Example 1. Let R be a semisimple ring. Then by Theorem 3, M = R(N)
is a δH R-module. Since R(N) ∼= R(N)⊕R(N) and R(N) 6= 0, M is not a gH
(Dedekind-finite) module (see [4, Corollary 1.4]).
The following lemma gives a source of examples of δH modules.
Lemma 5. Let M be a projective and semisimple R-module. Then M
is δH.
“adm-n4” — 2019/1/24 — 10:02 — page 177 — #27
S. E. Atani, M. Khoramdel, S. D. Pishhesari 177
Proof. If M be a projective and semisimple R-module, then M ≪δ M ,
by [12, Lemma 2.9] and so every surjective endomorphism of M has a
δ-small kernel.
In the following, it is shown that every δH R-module is gH if and only
if δ(R) = J(R).
Theorem 4. Let R be a ring. Then the following statements are equivalent:
(1) The class of δH R-modules coincide with the class of gH R-modules;
(2) Every projective δH R-module is gH;
(3) Every maximal right ideal of R is essential in RR;
(4) R has no non-zero semisimple projective R-module;
(5) δ(R) = J(R).
Proof. (1) ⇒ (2) Is clear.
(2) ⇒ (3) Let m be a maximal right ideal of R. It is clear that either
m is essential in RR or a direct summand of RR. If m is a direct summand
of RR, then M = (R/m)(N) is projective and semisimple. Hence M is δH
by Lemma 5. Therefore by (2), M is gH. As M ∼= M ⊕M , M = 0, by
[4, Theorem 1.1]. This is a contradiction, and so m is essential in RR.
(3) ⇒ (4) Is clear.
(4) ⇒ (1) Let M be a δH module and f : M → M⊕X an epimorphism.
Since M is δH, X is projective and semisimple by Theorem 1. Therefore
X = 0, by (4), and so M is gH, by [4, Theorem 1.1].
(3) ⇒ (5) Is clear.
(5) ⇒ (3) Let R be a ring such that δ(R) = J(R). If m is a maximal
right ideal of R such that m 6⊕ RR, say R = m ⊕ m
′ for some right
ideal m′ of R, then m
′ ⊆ Soc(R) ⊆ δ(R) ⊆ J(R) ⊆ m, a contradiction.
Therefore every maximal right ideal of R is essential in RR.
Lemma 6. Let R be a domain, which is not a division ring. Then δ(R) =
J(R).
Proof. Let x ∈ δ(R). Then xR ≪δ R. We show that xR ≪ R. Let
xR+K = R for some K 6 RR. By Lemma 1, there exists Y 6 xR such
that Y ⊕K = R. As R is a domain, Y = R or K = R. If Y = R, then
xR = R. Hence δ(R) = R, therefore R is semisimple by [18, Corollary 1.7].
Hence R is a division ring, a contradiction. Therefore K = R and so
xR ≪ R. It implies that δ(R) = J(R).
A direct sum of δH modules need not be a δH module, as the following
example shows.
“adm-n4” — 2019/1/24 — 10:02 — page 178 — #28
178 δ-Hopfian modules
Example 2. ([4, Remark 1.5], [10, Page 19, Exercise 18]) Let R be the
K-algebra generated over a field K by {s, t, u, v, w, x, y, z} with relations
sx+ uz = 1, sy + uw = 0, tx+ vz = 0 and ty + vw = 1.
Then R is a domain which is not a division ring. Hence by Lemma 6,
δ(R) = J(R). By [4, Remark 1.5],R is gH, however R2 is not gH. Therefore
R is δH, but R2 is not δH.
The next result gives a condition that a direct sum of two δH modules
is δH.
Proposition 3. Let M1 and M2 be two R-modules. If for every i ∈ {1, 2},
Mi is a fully invariant submodule of M = M1 ⊕M2, then M is δH if and
only if Mi is δH for each i ∈ {1, 2}.
Proof. The necessity is clear from Proposition 1. For the sufficiency, let
f = (fij) be a surjective endomorphism of M , where fij ∈ Hom(Mi,Mj)
and i, j ∈ {1, 2}. By assumption, Hom(Mi,Mj) = 0 for every i, j ∈ {1, 2}
with i 6= j. Since f is an epimorphism, fii is a surjective endomorphism
of Mi for each i ∈ {1, 2}. As Mi is δH for each i ∈ {1, 2}, Ker(fii) ≪δ Mi.
Since Ker(f) = Ker(f11)⊕Ker(f22), Ker(f) ≪δ M by Lemma 2(3). Hence
M is δH.
In the following example, it is shown that the δH property of a module
dose not inherit by its endomorphism ring.
Example 3. Let M be an infinite dimensional vector space over a division
ring K. Then by Theorem 3, M is δH. Since S ∼= S2 by [5, Example 5.16]
and S is not a semisimple ring, S = End(M) is not δH, by Theorem 1.
Theorem 5. Let M be a quasi-projective R-module. Then M is δH if
and only if M/N is δH for any small submodule N of M .
Proof. Let M be δH, N 6 M and f : M/N → M/N be an epimorphism.
Since M is quasi-projective, there exists a homomorphism g : M → M
such that πg = fπ, where π : M → M/N is the natural epimorphism. As
N ≪ M , g is an epimorphism, by [16, 19.2]. Therefore Ker(g) ≪δ M .
Since πg = fπ, g(N) 6 N and Ker(f) = (g−1(N))/N . As N ≪ M (and
so N ≪δ M) and g is an epimorphism, g−1(N) ≪δ M , by Theorem 1(2).
Therefore Ker(f) ≪δ M/N , by Lemma 2. Therefore M/N is δH. The
converse is clear by taking N = 0.
“adm-n4” — 2019/1/24 — 10:02 — page 179 — #29
S. E. Atani, M. Khoramdel, S. D. Pishhesari 179
Theorem 6. Let M be an R-module. If M satisfies a.c.c or d.c.c on non
δ-small submodules, then M is a δH module.
Proof. Let M be a module that satisfies a.c.c. on non δ-small submodules
and f : M → M an epimorphism. If Ker(f) is not δ-small in M , then
Ker(f) ⊆ Ker(f2) ⊆ Ker(f3) ⊆ . . . is an ascending chain of non δ-small
submodules of M . Hence there exists n > 1 such that Ker(fn) = Ker(fn+i)
for each i > 1. By a usual argument,Ker(f) = 0, a contradiction. Therefore
Ker(f) ≪δ M and so M is δH.
Assume that M satisfies d.c.c on non δ-small submodules and M is not
δH. Hence there exists an epimorphism f : M → M such that K = Ker(f)
is not a δ-small submodule of M . Therefore then each submodule L of
M , which contains K, is not a δ-small submodule of M . As M is not
δH, it is not Artinian. Hence M/K ∼= M is not Artinian and there is a
descending chain L1/K ⊃ L2/K ⊃ L3/K ⊃ . . . of submodules of M/K.
Thus L1 ⊃ L2 ⊃ L3 ⊃ . . . is a descending chain of non δ-small submodule
of M , a contradiction.
Proposition 4. Let R be a ring. If R/δ(R) is a semisimple ring, then
every finitely generated right R-module M is δH.
Proof. Assume that R/δ(R) is a semisimple ring, and M is a finitely
generated right R-module. Hence δ(M) = δ(R)M by [18, Theorem 1.8].
Therefore M/δ(M) is semisimple as an R/δ(R) − module, and so it is
semisimple as R-module. Therefore M/δ(M) is δH, by Theorem 6. As M
is finitely generated, δ(M) ≪δ M , and so M is δH, by Theorem 1(5).
The following result shows δH property is preserved under Morita
equivalences.
Theorem 7. δ-Hopfian is a Morita invariant property.
Proof. Let R and S be Morita equivalent rings with inverse category
equivalences α : Mod -R → Mod-S, β : Mod -S → Mod -R. Let M be
a δH R-module. We show that α(M) is a δH S-module. Assume that
φ : α(M) → α(M)⊕X be an S-module epimorphism where X is a right S-
module. Since any category equivalence preserves epimorphisms and direct
sums, we have β(φ) : βα(M) → βα(M)⊕β(X), as an epimorphism of right
R-modules. As βα(M) ∼= M , we have an epimorphism M → M ⊕β(X) of
R-modules. Therefore β(X) is semisimple and projective as an R-module,
by Theorem 1. Since any category equivalence preserves semisimple and
projective properties, X is semisimple and projective as an S-module.
Therefore α(M) is δH.
“adm-n4” — 2019/1/24 — 10:02 — page 180 — #30
180 δ-Hopfian modules
Corollary 2. Let n > 2. Then the following statements are equivalent
for a ring R:
(1) Every n-generated R-module is δH;
(2) Every cyclic Mn(R)-module is δH.
Proof. Let P = Rn and S = End(P ). Then, it is known that HomR(P,−):
NR → HomR(SPR, NR) defines a Morita equivalence between Mod-R and
Mod-S with the inverse equivalence −⊗S P : MS → M ⊗ P . Moreover, if
N is an n-generated R-module, then HomR(P,N) is a cyclic S-module
and for any cyclic S-module M , M ⊗S P is an n-generated R-module. By
Theorem 7, a Morita equivalence preserves the δH property of modules.
Therefore, every cyclic S-module is δH if and only if every n-generated
R-module is δH.
In the following, we characterize the rings R for which every finitely
generated free R-module is δH.
Corollary 3. Let R be a ring. Then the following statements are equiva-
lent:
(1) Every finitely generated free R-module is δH;
(2) Every finitely generated projective R-module is δH;
(3) Mn(R) is δH (as an Mn(R)-module) for each n > 1.
Proof. (1) ⇒ (2) It is clear from Proposition 1.
(2) ⇒ (1) It is clear.
(1) ⇔ (3) Let n be a positive integer and S = Mn(R). By the proof
of Corollary 2 and Theorem 7, if Rn is δH, then HomR(R
n, Rn) is δH as
an S-module. Conversely, if S is δH as an S-module, then S ⊗S Rn is δH
as an R-module.
2. Polynomial extensions of δ-Hopfian modules
Let M be an R-module. In this section we will briefly recall the
definitions of the modules M [X] and M [x]/(xn+1) from [13] and [17]. The
elements of M [X] are formal sums of the form m0 +m1x+ · · ·+mnx
n
with mi ∈ M and n ∈ N. We denote this sum by
∑n
i=0mix
i(m0x
0 is to
be understood as the element of M). Addition is defined by adding the
corresponding coefficients. The R[x]-module structure is given by
( k
∑
i=0
mix
i
)( t
∑
i=0
rix
i
)
=
k+t
∑
i=0
m′
ix
i,
“adm-n4” — 2019/1/24 — 10:02 — page 181 — #31
S. E. Atani, M. Khoramdel, S. D. Pishhesari 181
where m′
p =
∑
i+j=pmirj , rj ∈ R and mi ∈ M . Any nonzero element β
of M [x] can be written uniquely as
∑l
i=k mix
i with l > k > 0, mi ∈ M ,
mk 6= 0 and ml 6= 0. In this case, we refer to k as the order of β, l as
the degree of β, mk as the initial coefficient of β, and ml as the leading
coefficient of β.
Let n be any non-negative integer and
In+1 = {0} ∪ {β| 0 6= β ∈ R[x], order of β > n+ 1}.
Then In+1 is a two-sided ideal of R[x]. The quotient ring R[x]/In+1 will
be called the truncated polynomial ring, truncated at degree n+ 1. Since
R has an identity element, In+1 is the ideal generated by xn+1. Even when
R does not have an identity element, we will denote the ring R[x]/In+1
by R[x]/(xn+1). Any element of R[x]/(xn+1) can be uniquely written as
∑t
i=0 rix
i, with ri ∈ R. Let Dn+1 = {0}∪{β| 0 6= β ∈ M [x], order of β >
n+1}. Then Dn+1 is an R[x]-submodule of M [x]. Since In+1M [x] ⊆ Dn+1,
we can see that R[x]/(xn+1) acts on M [x]/Dn+1. We denote the module
M [x]/Dn+1 by M [x]/xn+1. The action of R[x]/(xn+1) on M [x]/xn+1 is
given by
( n
∑
i=0
mix
i
)( n
∑
i=0
rix
i
)
=
n
∑
i=0
m′
ix
i,
where m′
p =
∑
i+j=pmirj , rj ∈ R and mi ∈ M . Any nonzero element β
of M [x]/Dn+1 can be written uniquely as
∑n
i=k mix
i with n > k > 0,
mi ∈ M and mk 6= 0. In this case, k is called the order of β and mk the
initial coefficient of β.
Proposition 5. Let M be an R-module. If M [x] is δH as an R[x]-module,
then M is δH.
Proof. Let M [x] be δH as an R[x]-module and f a surjective endomor-
phism of M . Assume that Ker(f)+K = M , for some K 6 M , with M/K
singular. Define f̄ : M [x] → M [x] by f̄(
∑n
j=0mjx
j) =
∑n
j=0 f(mj)x
j . It
is easy to see that f̄ is a surjective endomorphism of M [x] and Ker(f̄) =
Ker(f)[x]. Hence Ker(f̄) +K[x] = M [x]. We show that M [x]/K[x] is a
singular R[x]-module. Let β = m0 +m1x+ · · ·+mnx
n ∈ M [x]. As M/K
is singular, for each 0 6 i 6 n, there exists Ii 6
ess R such that miIi ⊆ K.
Put I = ∩n
i=0Ii. Therefore I 6ess RR and βI ⊆ K[x]. We claim that
I[x] 6ess R[x]. Let α = r0+ r1x+ · · ·+ rtx
t ∈ R[x], where t ∈ N. If r0 6= 0,
then t0 ∈ R exists such that 0 6= r0t0 ∈ I (because I 6ess RR). Now, if
r1t0 6= 0, then there exists t1 ∈ R such that 0 6= r1t0t1 ∈ I. Continuing
“adm-n4” — 2019/1/24 — 10:02 — page 182 — #32
182 δ-Hopfian modules
this process, we get r ∈ R such that 0 6= αr ∈ I[x] and I[x] 6ess R[x]. As
βI ⊆ K[x], βI[x] ⊆ K[x]. Thus M [x]/K[x] is singular. Since M [x] is δH,
Ker(f̄) ≪δ M [x]. Therefore K[x] = M [x] and so M = K. This implies
that Ker(f) ≪δ M and M is δH.
The following example shows that the converse of proposition 5 is not
correct.
Example 4. Let R be a semisimple ring, where R[x] is not semisimple,
and M = R(N). As M ∼= M ⊕ M and M [x] ∼= M ⊗R R[x], we have
M [x] ∼= M [x]⊕M [x]. Since R is semisimple, M is δH, by Theorem 3. As
M [x] is not semisimple, M [x] is not δH, by Theorem 1.
Remark 1. Let M be an R-module and N a submodule of M [x]/(xn+1)
as an R[x]/(xn+1)-module, where n > 0. Define
Ni = {0} ∪ {initial coefficients of elements of order i in N}
for each 1 6 i 6 n. By [17], Ni 6 M and N0 ⊆ N1 ⊆ · · · ⊆ Nn.
Definition 3. Let M be an R-module and N a submodule of M [x]/(xn+1)
as an R[x]/(xn+1)-module, where n > 0. Then we say that N0 ⊆ N1 ⊆
· · · ⊆ Nn is the adjoint chain of N .
In the following, we show that for each submodule N of M [x]/(xn+1)
as an R[x]/(xn+1)-module, its adjoint chain plays an important role to
find its properties. By the definition of Ni (1 6 i 6 n), it is clear that Ni
is uniquely determined by N .
Lemma 7. Let N be a submodule of M [x]/(xn+1) as an R[x]/(xn+1)-
module, where n > 0. Then N 6ess M [x]/(xn+1) if and only if Nn 6ess
MR.
Proof. Let N 6ess M [x]/(xn+1) and 0 6= m ∈ M . Then there exists r0 +
r1x+ · · ·+rnx
n ∈ R[x]/(xn+1) such that 0 6= m(r0+r1x+ · · ·+rnx
n) ∈ N .
Let s be the order of m(r0+r1x+· · ·+rnx
n). Hence 0 6= mrsx
s ∈ Ns ⊆ Nn.
Therefore Nn 6ess M .
Conversely, assume that Nn 6ess M and msx
s + ms+1x
s+1 + · · · +
mnx
n ∈ M [x]/(xn+1) of order s. Since ms 6= 0 and Nn 6ess M , there
exists r ∈ R such that 0 6= msr ∈ Nn. Therefore 0 6= (msx
s +ms+1x
s+1 +
· · · +mnx
n)(rxn−s) = msrx
n. Clearly msrx
n ∈ N (by the definition of
Nn). Therefore N 6ess M [x]/(xn+1).
“adm-n4” — 2019/1/24 — 10:02 — page 183 — #33
S. E. Atani, M. Khoramdel, S. D. Pishhesari 183
Lemma 8. Let N be a submodule of M [x]/(xn+1) as an R[x]/(xn+1)-
module (n > 0) and N0 ⊆ N1 ⊆ · · · ⊆ Nn the adjoint chain of N . Then
N is a maximal submodule of M [x]/(xn+1) if and only if N0 is a maximal
submodule of M and Ni = M for each 1 6 i 6 n. Moreover, if N is a
maximal submodule of M [x]/(xn+1), then N = N0 +Mx+ · · ·+Mxn.
Proof. Let N be a maximal submodule of M [x]/(xn+1). Let N0 ⊆ N ′
0 for
some N ′
0 6 M . Set
N ′ = {m0 +m1x+ · · ·+mnx
n ∈ M [x]/(xn+1) : m0 ∈ N ′
0,mi ∈ M}.
It is clear that N ′ 6 M [x]/(xn+1) and N ⊆ N ′. Since N is maximal,
N = N ′ or N ′ = M [x]/(xn+1). Therefore N0 = N ′
0 or N ′
0 = M , where N ′
0
is the first component of the adjoint chain of N ′. This implies that N0 is
maximal in M . Also, maximality of N gives Ni = M for each 1 6 i 6 n.
Conversely, assume that N0 is a maximal submodule of M and Ni = M
for each 1 6 i 6 n. If there exists N ′ � M [x]/(xn+1) such that N ⊆ N ′,
then N0 ⊆ N ′
0 and Ni ⊆ N ′
i for each 1 6 i 6 n, where N ′
0 ⊆ N ′
1 ⊆ · · · ⊆ N ′
n
is the adjoint chain of N ′. As N ′ 6= M [x]/(xn+1), N0 = N ′
0 and N ′
i = M
for each 1 6 i 6 n. Therefore N = N ′ and N is a maximal submodule of
M [x]/(xn+1).
Now, it is clear that, if N is a maximal submodule of M [x]/(xn+1) ,
then N = N0 +Mx+ · · ·+Mxn.
Now, we are ready to determine the δ(M [x]/(xn+1)) for a module M .
Theorem 8. Let M be an R-module. Then δ(M [x]/(xn+1)) = Rad(M) +
Mx+Mx2 + · · ·+Mxn.
Proof. By Lemmas 7 and 8, every maximal submodule of M [x]/(xn+1) is
essential. Hence
δ(M [x]/(xn+1))
=
⋂
{N6M [x]/(xn+1) : (M [x]/(xn+1))/N is simple and singular}
=
⋂
{N : N is maximal in M [x]/(xn+1)}
= Rad(M) +Mx+Mx2 + · · ·+Mxn.
It is known that, ifM is anR-module andK ≪ M , thenK[x]/(xn+1) ≪
M [x]/(xn+1) as an R[x]/(xn+1)-module, by [17, Lemma 2.1]. However, it
is not true for δ-small submodules, as the following example shows.
“adm-n4” — 2019/1/24 — 10:02 — page 184 — #34
184 δ-Hopfian modules
Example 5. Let F be a field and R = T2(F ), the ring of upper triangular
matrices over F . Then δ(RR) =
(
0 F
0 F
)
and J(R) =
(
0 F
0 0
)
. Let
I =
(
0 0
0 F
)
. Then I ≪δ R, because I ⊆ δ(RR) and δ(RR) ≪δ R.
However I[x]/(xn+1) is not a δ-small right ideal of R[x]/(xn+1), because
I[x]/(xn+1) 6⊂ δ(R[x]/(xn+1)) = J(R) +Rx+ · · ·+Rxn, by Theorem 8.
In [17, Theorem 2.2], it is shown that, if M is a gH R-module, then
M [x]/(xn+1) is gH as an R[x]/(xn+1)-module, however it is not true that,
if M is a δH R-module, then M [x]/(xn+1) is δH as an R[x]/(xn+1)-module,
as the following example shows.
Example 6. Let R be a semisimple ring and M = R(N). Then M
is δH by Theorem 3. Define f : M → M by f((r1, r2, . . . , rn, . . . )) =
(r2, r3, . . . , rn, . . . ). Then f is an epimorphism and
Ker(f) = {(r, 0, 0, 0, . . . ) ∈ R(N) : r ∈ R}.
It is clear that α : M [x]/(xn+1) → M [x]/(xn+1) defined by
α
( n
∑
j=0
mjx
j
)
=
n
∑
j=0
f(mj)x
j
is an R[x]/(xn+1)-epimorphism and Ker(α) = (Ker(α))[x]/(xn+1). If
Ker(α) ≪δ M [x]/(xn+1), then
Ker(α) ⊆ δ(M [x]/(xn+1)) = Rad(M) +Mx+Mx2 + · · ·+Mxn
by Theorem 8. But Rad(M) = 0 and Ker(α) 6⊂ δ(M [x]/(xn+1)). Therefore
Ker(α) is not a δ-small submodule of M [x]/(xn+1) and M [x]/(xn+1) is
not a δH module.
Theorem 9. Let M be an R-module. If M [x]/(xn+1) is δH as an
R[x]/(xn+1)-module, then M is δH.
Proof. Assume that M [x]/(xn+1) is δH as an R[x]/(xn+1)-module and
f : M → M an R-epimorphism. Define α : M [x]/(xn+1) → M [x]/(xn+1)
by
α
( n
∑
j=0
mjx
j
)
=
n
∑
j=0
f(mj)x
j .
“adm-n4” — 2019/1/24 — 10:02 — page 185 — #35
S. E. Atani, M. Khoramdel, S. D. Pishhesari 185
Then α is an R[x]/(xn+1)-epimorphism and Ker(α) = (Ker(f))[x]/(xn+1).
We show that Ker(f) ≪δ M . Let H be a submodule of M such that
Ker(f) +H = M with M/H singular. Hence
M [x]/(xn+1) = (Ker(f))[x]/(xn+1) +H[x]/(xn+1).
We claim that M [x]/(xn+1)
H[x]/(xn+1)
is singular as R[x]/(xn+1)-module. Let
m = m0 +m1x+ · · ·+mnx
n ∈ M [x]/(xn+1).
For each 0 6 j 6 n, there exists Ij 6
ess R such that mjIj ⊆ H. Put I =
∩n
i=1Ij . Then I 6ess R and so I[x]/(xn+1) 6ess R[x]/(xn+1), by Lemma 7.
As mjI ⊆ H for each 0 6 j 6 n, m(I[x]/(xn+1)) ⊆ H[x]/(xn+1). There-
fore M [x]/(xn+1)
H[x]/(xn+1)
is singular. As Ker(α) ≪δ M [x]/(xn+1), H[x]/(xn+1) =
M [x]/(xn+1), and so H = M . Therefore Ker(f) ≪δ M and M is δH.
3. Triangular matrix extensions
Throughout this section T will denote a 2-by-2 generalized (or formal)
triangular matrix ring
(
S M
0 R
)
, where R and S are rings and M is an
(S,R)-bimodule.
Proposition 6. Assume that M is an (S,R)-bimodule, and T =
(
S M
0 R
)
.
Then δ(TT ) =
(
H M
0 δ(RR)
)
, where
H = δ(SS) ∩ {I : I is a maximal right ideal of S with annS(M) ⊆ I}.
Proof. By Lemma 4, every maximal essential right ideal of T has the form
(
S M
0 J
)
, where J is a maximal essential right ideal of R or
(
I M
0 R
)
,
where I is a maximal right ideal with I ∩ annS(M) 6ess (annS(M))S .
Therefore δ(TT ) =
(
K M
0 δ(RR)
)
, where
K = {I 6 S : I is a maximal right ideal of S
with I ∩ annS(M) 6ess (annS(M))S}.
We prove K = H. If annS(M) = 0, then it is clear that K = H = δ(SS).
Assume that annS(M) 6= 0. Let x ∈ K and I be a maximal right ideal
“adm-n4” — 2019/1/24 — 10:02 — page 186 — #36
186 δ-Hopfian modules
of S. If annS(M) ⊆ I, then annS(M) = I ∩ annS(M) 6ess (annS(M))S .
Therefore x ∈ K implies that x ∈ I. If I 6ess SS , then I ∩ annS(M) 6ess
(annS(M))S . Hence x ∈ I. Therefore x ∈ H and so K ⊆ H . For the reverse
of the inclusion, let x ∈ H . Let I be a maximal right ideal of S such that
I ∩ annS(M) 6ess (annS(M))S . If annS(M) ⊆ I, then x ∈ I. Assume
that annS(M) 6⊂ I. Hence annS(M)+ I = S and so S/I ∼= annS(M)/(I ∩
annS(M)). As annS(M) 6= 0 and annS(M)∩ I 6ess (annS(M))S , we have
annS(M)∩ I 6= 0 and S/I is singular. Therefore I 6ess SS . Thus x ∈ δ(S)
gives x ∈ I; hence K = H.
The next result gives a characterization for the δH condition for a
2-by-2 generalized triangular matrix ring.
Theorem 10. Assume that M is an (S,R)-bimodule, and T =
(
S M
0 R
)
.
Then the following statements are equivalent:
(1) TT is δH.
(2) (i) SS is δH and if a ∈ S has the right inverse b, then 1− ba ∈ I,
for each maximal right ideal of R with annS(M) ⊆ I.
(ii) RR is δH.
Proof. (1) ⇒ (2) Let a ∈ S have the right inverse b ∈ S. Then
(
a 0
0 1
)(
b 0
0 1
)
=
(
1 0
0 1
)
. Since T is δH,
(
1 0
0 1
)
−
(
b 0
0 1
)(
a 0
0 1
)
∈
δ(TT ), by Theorem 2. Thus
1−ba ∈ δ(SS)∩{I : I is a maximal right ideal of S with annS(M) ⊆ I},
by Proposition 6. Hence SS is δH, by Theorem 2 and 1− ba ∈ I, for each
maximal right ideal of S with annS(M) ⊆ I.
(ii) It is similar to the proof of (i).
(2) ⇒ (1) Let
(
a m
0 p
)(
b n
0 q
)
=
(
1 0
0 1
)
, where a, b ∈ S, p, q ∈ R
and m,n ∈ M . Hence ab = 1 and pq = 1. By (1) and Proposition 6,
(
1 0
0 1
)
−
(
b n
0 q
)(
a m
0 p
)
∈ δ(TT ). Hence by Theorem 2, TT is δH.
Theorem 11. Let T =
(
S M
0 R
)
, where M is an (S,R)-bimodule. If M
is a faithful left R-module, then TT is δH if and only if SS is Dedekind
finite and RR is δH.
“adm-n4” — 2019/1/24 — 10:02 — page 187 — #37
S. E. Atani, M. Khoramdel, S. D. Pishhesari 187
Proof. By Proposition 10, δ(TT ) =
(
H M
0 δ(RR)
)
, where
H = δ(SS) ∩ {I : I is a maximal right ideal of S with annS(M) ⊆ I}.
Since annS(M) = 0, H = J(S). Let TT be δH. If ab = 1 (a, b ∈ S), then
similar to the proof of Theorem 10, 1− ba ∈ J(S), and so ba = 1. This
implies that S is Dedekind-finite. Also, from Theorem 10, RR is δH. The
converse can be concluded from Theorem 10.
Since MR is always a faithful left S-module for S = End(MR), we have
the following corollary. It is known that an R-module M is Dedekind-finite
if and only if EndR(M) is a Dedekind-finite ring.
Corollary 4. Let T =
(
End(MR) M
0 R
)
. Then TT is δH if and only if
M is Dedekind-finite and RR is δH.
Theorem 12. Assume R is a ring. Then the following are equivalent:
(1) RR is Dedekind-finite;
(2) Tn(R) is δH, for every positive integer n.
Proof. (1) ⇒ (2) We proceed by induction on n. Note that Tn+1(R) =
(
R M
0 Tn(R)
)
, where M = (R,R, . . . , R) (n-tuple). For n = 2, if R is
Dedekind-finite, then T2(R) is δH, by Theorem 11. Now, assume that R is
Dedekind-finite and Tn(R) is δH. Hence by Theorem 11, Tn+1(R) is δH.
(2) ⇒ (1) It is clear from Theorem 11.
Theorem 13. Let R be a ring and U(R) the countably upper triangular
matrix ring over R. Then R is Dedekind-finite if and only if U(R) is δH.
Proof. It is clear that U(R) ∼=
(
R M
0 U(R)
)
, where M = (R,R, . . . ). Now,
the result is clear from Theorem 11.
Motivated by [1, Proposition 2.14], we have the following theorem.
Theorem 14. Let SMR be a nonzero (S,R)-bimodule such that Mn
R is
δH for all n > 1. Then either MR is semisimple and projective or one of
the rings R or S satisfies the rank condition.
“adm-n4” — 2019/1/24 — 10:02 — page 188 — #38
188 δ-Hopfian modules
Proof. Assume that MR is not projective or semisimple. Let T =
(
S M
0 R
)
and I =
(
0 M
0 0
)
. Since MR is not projective or semisimple, IT is not
projective or semisimple. By hypothesis, we can conclude that InT is δH for
each n > 1. Now we will show that the ring T satisfies the rank condition.
Assume that T does not satisfy the rank condition and f : T p → T q is
an epimorphism with q > p. Thus f(Ip) = f(T pI) = f(T p)I = T qI = Iq.
Hence f : Ip → Ip⊕Iq−p is an epimorphism. Since InT is δH for each n > 1,
Iq−p is semisimple and projective as T -module, by Theorem 1. But then
I should be projective and semisimple, which is not. Hence T satisfies
the rank condition. Therefore one of the rings R or S satisfies the rank
condition, by [6, Proposition 4.1].
Open Problems. (1) What is the structure of rings whose finitely gen-
erated right modules are δH?
(2) Does Theorem 5 hold for δ-small submodules? (That is, let M be
a quasi-projective R-module. Then M is δH if and only if so is M/N for
any δ-small submodule N of M).
Acknowledgment
The authors express their deep gratitude to the referee for her/his
helpful suggestions for the improvement of this work.
References
[1] Sh. Asgari, A. Haghany, and M. R. Vedadi, Quasi Co-Hopfian Modules and
Applications, Comm Algebra, 36(2008), 1801–1816.
[2] G. Baumslag, Hopficity and abelian groups, in: J. Irwin, E.A. Walker (Eds.), Topics
in Abelian Groups, Scott Foresmann and Company, 1963, 331–335.
[3] G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Generalized triangular matrix rings
and the fully invariant extending property, Rocky Mountain J. Math., 32(2002),
1299–1319.
[4] A. Ghorbani, A. Haghany, Generalized Hopfian modules, J. Algebra, 255(2) (2002),
324–341.
[5] K.R. Goodearl, Von Neumann Regular Rings, Monographs and studies in mathe-
matics, 1979.
[6] A. Haghany, K. Varadarajan, IBN and related properties for rings. Acta Math.
Hung. 93 (2002), 251–261.
[7] A. Haghany, M.R. Vedadi, Modules whose injective endomorphisms are essential,
J. Algebra 243 (2001) 765–779.
“adm-n4” — 2019/1/24 — 10:02 — page 189 — #39
S. E. Atani, M. Khoramdel, S. D. Pishhesari 189
[8] V. Hiremath, Hopfian rings and Hopfian modules, Indian J. Pure Appl. Math. 17
(1986) 895–900.
[9] I. Kaplansky, Rings of Operators. New York-Amsterdam: W. A. Benjamin, Inc.
(1968).
[10] T.Y. Lam, Lectures on modules and rings, New York: Springer-Verlag, 1998.
[11] R. Triback, Finitely generated δ-supplemented modules are amply δ-supplemented,
Bull. Aust. Math. Soc. 86 (2012), 430–439.
[12] R. Triback, On δ-Local Modules and Amply δ-Supplemented Modules, Journal of
Algebra and Its Applications Vol. 12, No. 2 (2013) 1250144.
[13] K. Varadarajan, A generalization of Hilbert’s basis theorem. Comm. Algebra 10(20)
(1982), 2191–2204.
[14] K. Varadarajan, Hopfian and co-Hopfian objects. Publicacions Matemàtiques
36(1992), 293–317.
[15] K. Varadarajan, On Hopfian rings. Acta Math. Hungar. 83(1-2) (1999), 17–26.
[16] R. Wisbauer, Foundations of Module and Ring Theory. Gordon and Breach Science
Publishers, 1991.
[17] G. Yang, Z. Liu, Notes on Generalized Hopfian and Weakly Co-Hopfian Modules,
Comm. Algebra, 38(2010), 3556–3566.
[18] Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra
Colloquium, 7 (3) (2000), 305–318.
Contact information
S. Ebrahimi Atani,
M. Khoramdel,
S. Dolati Pishhesari
Department of Mathematics, University of
Guilan, P.O.Box 1914, Rasht, Iran
E-Mail(s): ebrahimi@guilan.ac.ir,
mehdikhoramdel@gmail.com,
saboura−dolati@yahoo.com
Received by the editors: 15.12.2016
and in final form 18.10.2018.
|