Modules with Unique Closure Relative to a Torsion Theory. III
We continue the study of modules over a general ring R whose submodules have a unique closure relative to a hereditary torsion theory on Mod-R. It is proved that, for a given ring R and a hereditary torsion theory τ on Mod-R, every submodule of every right R-module has a unique closure with respect...
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2014
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508132839522304 |
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| author | Doğruöz, S. Harmanci, A. Smith, P. F. Догруоз, С. Харманці, А. Сміт, П. Ф. |
| author_facet | Doğruöz, S. Harmanci, A. Smith, P. F. Догруоз, С. Харманці, А. Сміт, П. Ф. |
| author_sort | Doğruöz, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2019-12-05T10:25:58Z |
| description | We continue the study of modules over a general ring R whose submodules have a unique closure relative to a hereditary torsion theory on Mod-R. It is proved that, for a given ring R and a hereditary torsion theory τ on Mod-R, every submodule of every right R-module has a unique closure with respect to τ if and only if τ is generated by projective simple right R-modules. In particular, a ring R is a right Kasch ring if and only if every submodule of every right R-module has a unique closure with respect to the Lambek torsion theory. |
| first_indexed | 2026-03-24T02:20:21Z |
| format | Article |
| fulltext |
UDC 512.5
S. Doğruöz (Adnan Menderes Univ., Aydin, Turkey),
A. Harmanci (Hacettepe Univ., Ankara, Turkey),
P. F. Smith (Univ. Glasgow, Scotland, UK)
MODULES WITH UNIQUE CLOSURE RELATIVE TO A TORSION THEORY. III
МОДУЛI З ЄДИНИМ ЗАМИКАННЯМ ВIДНОСНО ТЕОРIЇ СКРУТУ. III
We continue the study of modules over a general ring R whose submodules have a unique closure relative to a hereditary
torsion theory on Mod-R. It is proved that, for a given ring R and a hereditary torsion theory τ on Mod-R, every
submodule of every right R-module has a unique closure with respect to τ if and only if τ is generated by projective simple
right R-modules. In particular, a ring R is a right Kasch ring if and only if every submodule of every right R-module has
a unique closure with respect to the Lambek torsion theory.
Продовжено вивчення модулiв над загальним кiльцемR, субмодулi якого мають єдине замикання вiдносно спадкової
теорiї скруту на Mod-R. Доведено, що для заданих кiльця R та спадкової теорiї скруту τ на Mod-R кожний
субмодуль кожного правого R-модуля має єдине замикання вiдносно τ тодi i тiльки тодi, коли τ породжується
проективними простими правими R-модулями. Зокрема, кiльце R є правим кiльцем Каша тодi i тiльки тодi, коли
кожний субмодуль кожного правого R-модуля має єдине замикання вiдносно теорiї скруту за Ламбеком.
1. Introduction. In this note all rings are associative with identity and all modules are unitary right
modules. This paper is a continuation of [1] and [2], and any unexplained terms can be found in
[1, 3, 4]. Let R be a ring. All torsion theories on Mod-R, the category of all right R-modules, will
be hereditary. For the Goldie torsion theory and terminology for torsion theories in general and any
unexplained terminology see also [5, 6, 9].
Let R be any ring and let τ be a torsion theory on Mod-R. Given an R-module M, τ(M) will
denote the τ -torsion submodule of M. A submodule L of M is called τ -essential provided L is an
essential submodule of M and M/L is a τ -torsion module. In addition, a submodule K of M is
called τ -closed in M provided K has no proper τ -essential extension in M, i.e., if K is a τ -essential
submodule in a submodule L of M, then K = L. Note that if K is a submodule of M such that
either M/K is τ -torsion-free or K is a closed submodule of M then K is a τ -closed submodule of
M. Given a submodule N of M, by a τ -closure of N in M we mean a τ -closed submodule K of
M containing N such that N is τ -essential in K. The module M is called a τ -UC-module provided
every submodule has a unique τ -closure in M.
In [1, 2] we investigate, for a general torsion theory τ, when a submodule of a general R-module
M has a unique τ -closure and when the module M is τ -UC. In this paper we are interested when
a ring R has the property that every (right) R-module is τ -UC, for a given torsion theory τ. One
consequence of [10] (Theorem (1) ⇔ (16)) is that the ring R is semiprime Artinian if and only if
every (right) R-module is UC. The Goldie torsion theory will be denoted by τG. It is pointed out in
[1, p. 232] that every τG-UC-module is UC, and conversely. Thus R is semiprime Artinian if and
only if every module is τG-UC. Recall that if τ and ρ are torsion theories on Mod-R then τ ≤ ρ
provided every τ -torsion module is ρ-torsion. It is proved in [1] (Proposition 3.6) that if τ and ρ are
torsion theories on Mod-R such that τ ≤ ρ then every ρ-UC-module is τ -UC and in particular every
UC-module is a τ -UC-module. We have proved our first result.
c© S. DOĞRUÖZ, A. HARMANCI, P. F. SMITH, 2014
922 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
MODULES WITH UNIQUE CLOSURE RELATIVE TO A TORSION THEORY. III 923
Proposition 1.1. Let R be a ring and let τ be any torsion theory on Mod-R. Then R semiprime
Artinian implies that every R-module is τ -UC. Moreover the converse holds in case τG ≤ τ.
LetM be anR-module. For any nonempty subsetX ofM, annR(X) will denote {r ∈ R : xr = 0
for all x ∈ X} and for any nonempty subset Y of R, annM (Y ) will denote {m ∈ M : my = 0 for
all y ∈ Y }. In case X = {x} and Y = {y} we shall write annR(x) and annM (y) for annR(X) and
annM (Y ), respectively. Now we prove the following result.
Theorem 1.1. Let R be any ring. Then the following statements are equivalent for a torsion
theory τ on Mod-R.
(1) Every right R-module is τ -UC.
(2) For every right R-module X and τ -essential submodule Y of X, then Y = X.
(3) Every τ -torsion right R-module is projective.
(4) If E is an essential right ideal of R then the R-module R/E is τ -torsion-free.
(5) If a right ideal E of R is τ -essential in RR, then E = R.
(6) Every singular right R-module is τ -torsion-free.
(7) Every proper submodule of any module is τ -closed.
Proof. (1) ⇒ (2). Let Y be a τ -essential submodule of X. Then X ⊕ (X/Y ) is not τ -UC if
Y 6= X by [1] (Lemma 3.2). By hypothesis (1), Y = X.
(2)⇒ (3). Let M be any τ -torsion R-module. There exists a free R-module F and a submodule
K of F such that M ∼= F/K. Let L be a submodule of F maximal with respect to the property
K ∩ L = 0 (Zorn’s lemma). Then K ⊆ K ⊕ L and K ⊕ L is an essential submodule of F. Since
F/K is τ -torsion it follows that K ⊕ L is a τ -essential submodule of F and hence F = K ⊕ L and
M ∼= L is projective.
(3) ⇒ (4). Let E be an essential right ideal of R. There exists a submodule E′ of RR with
E ⊆ E′ such that E′/E is τ -torsion and R/E′ τ -torsion-free. By (3), E′/E is projective and hence
E′ = E. Thus R/E is τ -torsion-free.
(4) ⇒ (5). Let E be a τ -essential submodule of RR. Then E is essential in R and R/E is
τ -torsion. By (4), E = R.
(5) ⇒ (1). Let M be an R-module. Suppose M is not τ -UC. Then there exists a submodule
X of M and a proper τ -essential submodule Y of X such that the R-module X ⊕ (X/Y ) embeds
in M (see [1], Lemma 3.2). Let x ∈ X \ Y. Then xR ⊕ (xR + Y )/Y can be embedded in M,
i.e., xR ⊕ (xR/(xR ∩ Y )) embeds in M and xR ∩ Y is τ -essential in xR. If A = annR(x), then
xR ∼= R/A and xR∩ Y ∼= B/A where B/A is τ -essential in R/A. Then B is τ -essential in RR. By
(5), B = R a contradiction. Therefore M is a τ -UC-module.
(4) ⇒ (6). Let M be a singular R-module. Let m ∈ M. Then mR ∼= R/E for some essential
right ideal E of R. By (4), mR is a τ -torsion-free module. Hence M is τ -torsion-free.
(6) ⇒ (7). Let N be a proper submodule of an R-module M and let K be a τ -closure of N
in M. Then N is τ -essential in K and hence K/N is singular and so τ -torsion. But by hypothesis
K/N is τ -torsion-free. Thus N = K and hence N is τ -closed in M.
(7)⇒ (6). Let N be a singular R-module. Let u ∈ τ(N). Then uF = 0 for some essential right
ideal F of R. Note that R/F ∼= uR ⊆ τ(N) and hence F is τ -essential in R. By (5) F = R and
hence u = 0. It follows that N is τ -torsion-free.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
924 S. DOĞRUÖZ, A. HARMANCI, P. F. SMITH
(6)⇒ (4). Clear.
Theorem 1.1 is proved.
Corollary 1.1. If every R-module is τ -UC, then every τ -torsion R-module is semisimple.
Proof. Assume that every R-module is τ -UC. Then every singular right R-module is τ -torsion
free by Theorem 1.1 (6). Let M be a torsion R-module and let N be a submodule of M. By Zorn’s
lemma there exists a submodule K of M such that N ⊕K is essential in M. Then M/(N ⊕K) is
singular. By hypothesis M/(N ⊕K) is τ -torsion free, and it is also τ -torsion. Thus M = N ⊕K.
It follows that M is semisimple.
Corollary 1.1 is proved.
The converse of Corollary 1.1 need not be true in general.
Example 1.1. Let R be a commutative ring and let P be a maximal ideal of R such that P = P 2
and annR(P ) = 0. Let τ be the hereditary torsion theory generated by R/P. Then every τ -torsion
module is semisimple but not every R-module is τ -UC.
Proof. If R/P is a projective R-module then R = P ⊕ I for some ideal I of R. Then IP = 0
and hence I = 0, a contradiction. Thus R/P is not projective. It follows that P is an essential ideal
of R and hence R/P is a nonzero singular module which is τ -torsion and hence not τ -torsion-free.
Thus not every R-module is τ -UC by Theorem 1.1.
Let X be a τ -torsion module. Let S = {x ∈ X : Px = 0}. Suppose X 6= S. Then there
exists an R-submodule T of X such that S ⊂ T ⊆ X and T/S ∼= R/P. Then PT ⊆ S and hence
PT = P 2T = P (PT ) ⊆ PS = 0. Thus T ⊆ S, a contradiction. It follows that X = S and so X is
semisimple.
Example 1.2. Let p be any prime integer, let F be a field of characteristic p > 0 and let G be
the Prüfer p-group. Then the group algebra R = F [G] has augmentation ideal
P =
∑
x∈G
R(x− 1)
which satisfies P = P 2 and annR(P ) = 0.
Proof. By [8] (Lemma 3.1.2), annR(P ) = 0. Let x ∈ G. Then x = yp for some y ∈ G and
hence x− 1 = yp − 1 = (y − 1)p ∈ P 2. It follows that P = P 2.
Corollary 1.2. LetR be a ring such that every singular rightR-module is τ -torsion free, for some
torsion theory τ. ThenM/Soc(MR) is τ -torsion- free and τ(M) ⊆ Soc(MR) for every R-moduleM.
Proof. Let M be any R-module and let Ni (i ∈ I) denote the collection of essential submodules
of M. For each i ∈ I, M/Ni is τ -torsion-free. Thus
∏
i∈I
(M/Ni) is τ -torsion-free. Because
M/Soc(M) is isomorphic to a submodule of
∏
i∈I
(M/Ni), it is τ -torsion free. The last part clearly
follows.
Corollary 1.2 is proved.
Let R be any ring. Let τ(RR) =
⊕
i∈I Ui where Ui is a simple module for each i ∈ I. Choose
J ⊆ I such that Uj � Uk if j 6= k in J and for each i ∈ I there exists j ∈ J such that Ui ∼= Uj . Then
the torsion class of a torsion theory τ is generated by {Uj : j ∈ J}, i.e., X belongs to the torsion
class of τ ⇔ X =
⊕
λ∈Λ Vλ where for every λ ∈ Λ, Vλ ∼= Uj for some j ∈ J.
Now we characterize the torsion theories which satisfy the equivalent conditions in Theorem 1.1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
MODULES WITH UNIQUE CLOSURE RELATIVE TO A TORSION THEORY. III 925
Theorem 1.2. Let τ be a torsion theory on Mod-R. Then every right R-module is τ -UC if and
only if τ is generated by a collection S of projective simple R-modules.
Proof. Assume that every R-module is τ -UC. By Corollary 1.1, every τ -torsion module is
semisimple. Let S denote the set of representatives of τ -torsion simple modules. Let V ∈ S. Then
V ∼= R/P for some maximal right ideal P of R. Hence P is not essential. Otherwise R/P is
singular, so it is τ -torsion-free by Theorem 1.1. But since R/P is τ -torsion, this is a contradiction.
Therefore R = P ⊕X for some projective simple τ -torsion right ideal X. Then V ∼= X and so S
consists of all projective simple τ -torsion R-modules.
Conversely, let T be the collection of modules of the form
⊕
i∈IWi where for each i ∈ I there
exists Vi ∈ S such that Wi
∼= Vi. Then T is closed under submodules, homomorphic images and
direct sums. To complete the proof we show T is closed under extensions by short exact sequences.
Let X be an R-module and Y a submodule of X with Y ∈ T and X/Y ∈ T . Then X/Y is
projective, being a direct sum of projective simple modules. Hence Y is direct summand of X.
Therefore X ∼= Y ⊕ (X/Y ) ∈ T .
Theorem 1.2 is proved.
Let I be an idempotent ideal of a ring R. Then τI will denote the torsion theory whose torsion
modules are the R-modules X such that XI = 0. As an application of Theorem 1.2 we now
characterize the idempotent ideals I such that every R-module is a τI -UC-module.
Corollary 1.3. Let I be an idempotent ideal of a ring R. Then every R-module is τI -UC if and
only if I = eR for some idempotent element e of R and the ring R/I is semiprime Artinian.
Proof. Suppose first that every R-module is τI -UC. Let A be a right ideal of R maximal with
respect to the property that I ∩ A = 0. It is well known that I ⊕ A is an essential right ideal of R
and hence I ⊕ A is a τI -essential right ideal of R because (R/(I ⊕ A))I = 0. By Theorem 1.1 (5),
R = I ⊕ A. It follows that I = eR for some e = e2 ∈ R. Let E be a right ideal of R containing I
such that E/I is an essential right ideal of R/I. Then E is an essential right ideal of R and clearly E
is τI -essential in R. Again using Theorem 1.1, E = R. Thus the ring R/I does not contain a proper
essential right ideal and hence R/I is a semiprime Artinian ring. Conversely, suppose that I = eR
for some idempotent e ∈ R such that the ring R/I is semiprime Artinian. Let F be a right ideal of
R such that F is τI -essential in RR. Because R/F is τI -torsion, we have (R/F )I = 0 and hence
I ⊆ F. Next F is essential in RR and I is closed in RR so that F/I is an essential right ideal of
R/I by [4, p. 6]. But R/I is semiprime Artinian implies that F/I = R/I and hence F = R. By
Theorem 1.1, every R-module is τI -UC.
Corollary 1.3 is proved.
2. Further results. Let R be a ring and M an R-module. Given a submodule L of M and an
element m ∈ M then (L : m) will denote the set of elements r ∈ R such that mr ∈ L. Note that
(L : m) is a right ideal of R.
Lemma 2.1. Let R be any ring. The following conditions are equivalent for torsion theories
ρ, τ on Mod-R.
(1) Every ρ-essential right ideal of R is a τ -essential right ideal of R.
(2) For every R-module M, every ρ-essential submodule of M is a τ -essential submodule of M.
Proof. (2)⇒ (1). Apply (2) in case M = R.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
926 S. DOĞRUÖZ, A. HARMANCI, P. F. SMITH
(1) ⇒ (2). Let M be an R-module and let L be a ρ-essential submodule of M. Then L is an
essential submodule of M and M/L is ρ-torsion. Let m ∈ M and let A = (L : m). Then A is
an essential right ideal of R such that R/A ∼= (mR + L)/L so that R/A is ρ-torsion. Thus A is
a ρ-essential right ideal of R. In particular, R/A is τ -torsion and hence so too is (mR + L)/L. It
follows that M/L is τ -torsion and L is a τ -essential submodule of M.
Lemma 2.1 is proved.
Note the following result.
Proposition 2.1. Let R be any ring. Consider the following conditions for torsion theories ρ, τ
on Mod-R.
(1) ρ ≤ τ.
(2) Every ρ-essential right ideal of R is a τ -essential right ideal of R.
(3) Every τ -UC R-module is ρ-UC.
Then (1) ⇒ (2) ⇒ (3).
Proof. (1)⇒ (2). Clear.
(2) ⇒ (3). Let M be an R-module which is not ρ-UC. By [1] (Theorem 3.4) there exist an
R-module X and a proper ρ-essential submodule Y of X such that X ⊕ (X/Y ) embeds in M. Now
(2) and Lemma 2.1 show that Y is a τ -essential submodule of X. Applying [1] (Theorem 3.4) again
we see that M is not a τ -UC-module.
Proposition 2.1 is proved.
Corollary 2.1. LetR be any ring and let τ be a torsion theory onMod-R. Consider the following
conditions.
(1) τG ≤ τ.
(2) Every essential right ideal of R is a τ -essential right ideal of R.
(3) Every τ -UC R-module is a UC-module.
Then (1) ⇔ (2) ⇒ (3).
Proof. (1)⇒ (2) ⇒ (3). By Proposition 2.1.
(2) ⇒ (1). Let M be a nonzero singular R-module. Let m ∈M. Then annR(m) is an essential,
and hence τ -essential, right ideal of R. Then mR ∼= R/ annR(m) is τ -torsion. It follows that M is
τ -torsion. Thus τG ≤ τ.
Corollary 2.1 is proved.
Next we give criteria for τG ≤ τ.
Proposition 2.2. Let R be any ring and let τ be a torsion theory on Mod-R. Then the following
statements are equivalent.
(1) τG ≤ τ.
(2) For any R-module M and any submodule N of M there exists a submodule K of M such
that N ∩K = 0 and N ⊕K is τ -essential in M.
(3) For any right ideal A of R there exists a right ideal B of R such that A∩B = 0 and A⊕B
is τ -essential in RR.
(4) Every singular R-module is τ -torsion.
Proof. (1) ⇒ (2). Assume that τG ≤ τ. Let N be a submodule of M. By Zorn’s Lemma there
exists a submoduleK ofM maximal with respect to the propertyN∩K = 0. ThenN⊕K is essential
in M. Hence M/(N ⊕K) is singular. By assumption M/(N ⊕K) is τ -torsion and therefore N ⊕K
is τ -essential in M.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
MODULES WITH UNIQUE CLOSURE RELATIVE TO A TORSION THEORY. III 927
(2) ⇒ (3). Clear.
(3) ⇒ (4). Let M be any singular module. For any x ∈ M, xR ∼= R/ annR(x) where annR(x)
is an essential right ideal of R. Now R/ annR(x), and hence xR, is τ -torsion by (3). Therefore M
is τ -torsion. Thus every singular module is τ -torsion.
(4)⇒ (1). Let X be any τG-torsion module. There exists a submodule Y of X such that both Y
and X/Y are singular. By (4) both Y and X/Y are τ -torsion. It follows that X is τ -torsion. Thus
τG ≤ τ.
Proposition 2.2 is proved.
Next note that in Proposition 2.1, in general (2) and (3) do not imply (1) as the following example
shows.
Example 2.1. There are rings R and torsion theories ρ and τ on Mod-R such that every ρ-
essential right ideal of R is a τ -essential right ideal of R (and hence every τ -UC-module is ρ-UC)
but it is not the case that ρ ≤ τ.
Proof. Let R =
[
F F
0 F
]
be the ring of 2 × 2 upper triangular matrices over a field F. The
right ideals of R are 0, I1 =
[
F F
0 0
]
, I2 =
[
0 F
0 F
]
, I(x,y) =
{[
0 xc
0 yc
] ∣∣∣∣ c ∈ F}, for some
x, y ∈ F and R. Let τI1 = {N ∈ Mod-R | NI1 = 0} and τI2 = {N ∈ Mod-R | NI2 = 0} denote
the torsion theories determined by the idempotent ideals I1 and I2, respectively. The ring R has I2
as its only essential right ideal. Since (R/I2)I2 = 0, I2 is a τI2-essential right ideal of R. On the
other hand R does not have any τI1-essential right ideal except R itself. Hence every τI1-essential
right ideal of R is a τI2-essential right ideal of R. Since I(0,1)I1 = 0 and I(0,1)I2 = I(0,1) 6= 0, I(0,1)
is a τI1-torsion but not τI2-torsion module. Hence τI1 � τI2 .
Example 2.1 is completed.
Combining Proposition 2.1 and Example 2.1 we see that in general (3) does not imply (1) in
Proposition 2.1. We do not know whether (3) implies (2) in Corollary 2.1. In some situations (2)
does imply (1) in Proposition 2.1. For example, we have the following result.
Proposition 2.3. Let R be a ring and let ρ, τ be torsion theories on Mod-R such that ρ ≤ τG.
Then ρ ≤ τ if and only if every ρ-essential right ideal of R is a τ -essential right ideal of R.
Proof. The necessity follows by Proposition 2.1. Conversely, suppose that every ρ-essential right
ideal of R is a τ -essential right ideal of R. Let M be a ρ-torsion module. There exists a submodule
N of M such that both N and M/N are singular. Let m ∈ N. Because N is singular, annR(m)
is an essential right ideal of R. Since ρ is a hereditary torsion theory mR is ρ-torsion and thus
R/ annR(m) is also ρ-torsion since mR ∼= R/ annR(m). This implies that annR(m) is a ρ-essential
right ideal of R and therefore a τ -essential right ideal of R. Hence R/ annR(m) is τ -torsion, i.e.,
mR is τ -torsion for all m ∈ N, and hence N is τ -torsion. Similarly, M/N is τ -torsion. Thus M is
τ -torsion. It follows that ρ ≤ τ.
Proposition 2.3 is proved.
Let τ be any torsion theory on Mod-R. For any R-module M, Zτ (M) will denote the set of
elements m in M such that mE = 0 for some τ -essential right ideal E of R. Note that Zτ (M) is a
submodule of the singular submodule Z(M) of M.
Theorem 2.1. Let R be a ring and let τ be a hereditary torsion theory on Mod-R such that
Zτ (RR) = 0. Then τG ≤ τ if and only if every τ -UC-module is UC.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
928 S. DOĞRUÖZ, A. HARMANCI, P. F. SMITH
Proof. The necessity follows by Proposition 2.1. Conversely, suppose that every τ -UC-module
is a UC-module. Let E be any essential right ideal of R. Suppose that R/E is not τ -torsion. Then
there exists a proper right ideal F of R such that E ⊆ F and R/F is a τ -torsion-free R-module. Let
M denote the R-module R⊕ (R/F ). Note that Zτ (RR) = 0 and Zτ (R/F ) = 0 so that Zτ (M) = 0.
By [1] (Corollary 3.5), M is a τ -UC-module. However, since F is a proper essential right ideal
of R, M is not a UC-module by [10] (Theorem), a contradiction. Thus the R-module R/E is
τ -torsion for every essential right ideal E of R. It follows that every singular module is τ -torsion.
By Proposition 2.2, τG ≤ τ.
Theorem 2.1 is proved.
3. The Lambek torsion theory. Let R be a ring. For any R-module M, E(M) will denote
the injective hull of M. Let τL denote the Lambek torsion theory on Mod-R. Recall that τL is the
(hereditary) torsion theory on Mod-R whose torsion class consists of all R-modules M such that
Hom(M,E(RR)) = 0. For basic facts about the Lambek torsion theory see [5, 9]. Recall that the
ring R is called right nonsingular provided RR is τG-torsion-free.
Lemma 3.1. Let R be any ring. Then:
(1) τL ≤ τG;
(2) τL = τG if and only if R is right nonsingular.
Proof. (1) By [9] (Ch. VI, Corollary 6.5).
(2) By [9] (Ch. VI, Proposition 6.7 and Corollary 6.8).
Lemma 3.1 is proved.
Lemma 3.2. Let E = E(RR). Then an R-module M is τL-torsion if and only if
annE(annR(m)) = 0 for all m ∈M.
Proof. Let m ∈ M and let e ∈ annE(annR(m)). Define a mapping ϕ : mR → E by ϕ(mr) =
= er (r ∈ R). It is easy to check that ϕ is well-defined and an R-homomorphism. Moreover, for
every homomorphism θ : mR → E we have θ(m) ∈ annE(annR(m)). Because E is injective, the
result follows.
Lemma 3.2 is proved.
Corollary 3.1. A submodule N of an R-module M is a τL-essential submodule of M if and
only if
(a) N is an essential submodule of M, and
(b) annE(N : m) = 0 for all m ∈M, where E = E(RR).
Proof. The submodule N is a τL-essential submodule of M if and only if (a) holds and M/N
is τL-torsion. Apply Lemma 3.2.
Corollary 3.1 is proved.
Theorem 3.1. A ring R is right nonsingular if and only if every τL-UC-module is UC.
Proof. The necessity follows by Lemma 3.1 (2). Conversely, suppose that every τL-UC-module
is UC. We know that τL ≤ τG. Now we show that ZτL(RR) = 0. Suppose that x ∈ ZτL(RR).
Then there exists a τL-essential right ideal I of R such that xI = 0. Thus I is essential in RR and
R/I is τL-torsion. Therefore HomR(R/I,E(RR)) = 0. Define a mapping θ : R/I → E(RR) by
θ(r + I) = xr for all r ∈ R. Since xI = 0, θ is well-defined and is clearly an R-homomorphism.
Thus x = 0. It follows that ZτL(RR) = 0. By Theorem 2.1 τG ≤ τL. We have proved that τG = τL.
Finally Lemma 3.1 gives that R is right nonsingular.
Theorem 3.1 is proved.
Let R be a ring. Recall that R is called a right Kasch ring if every simple right module
embeds in R. Among examples of right Kasch rings are quasi-Frobenius rings (see, for example, [7],
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
MODULES WITH UNIQUE CLOSURE RELATIVE TO A TORSION THEORY. III 929
Theorem 3.4) and, more generally, right pseudo-Frobenius rings (see, for example, [7], Lemma 1.42
and Theorem 1.57). This brings us to our final theorem.
Theorem 3.2. The following statements are equivalent for a ring R.
(1) R is a right Kasch ring.
(2) Every τL-torsion module is zero.
(3) Every module is a τL-UC-module.
Proof. (1) ⇒ (2). Let R be a right Kasch ring and let MR 6= 0. For 0 6= m ∈ M, there
exists a maximal submodule K of mR. Then there is an embedding ϕ : mR/K → R and an
embedding ι : R → E(RR). By the injectivity E(RR), the homomorphism ιϕ extends to a nonzero
homomorphism θ : M/K → E(RR). If π : M →M/K denotes the natural epimorphism, then θπ is
a nonzero homomorphism from M to E(RR). Hence HomR(M,E(RR)) 6= 0.
(2) ⇒ (3). By Theorem 1.1.
(3) ⇒ (1). Let V be any simple R-module. Then V ∼= R/P for some maximal right ideal P of
R. If P is not an essential right ideal of R then there exists a nonzero right ideal U of R such that
P ∩ U = 0. In this case, R = P ⊕ U and hence V ∼= R/P ∼= U. Now suppose that P is an essential
right ideal of R. Because P 6= R, (3) combined with Theorem 1.1 gives that R/P is not τL-torsion.
Thus there exists a nonzero homomorphism α : V → E(RR). It follows that V ∼= ϕ(V ) ⊆ R, because
R is essential in E(RR). We have proved that R is a right Kasch ring.
Theorem 3.2 is proved.
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Conf. (May 1992). – Singapore: World Sci., 1992. – P. 302 – 313.
Received 18.07.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
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| id | umjimathkievua-article-2188 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:21Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/51/9aae8429fd89e34a9fac4e811873e151.pdf |
| spelling | umjimathkievua-article-21882019-12-05T10:25:58Z Modules with Unique Closure Relative to a Torsion Theory. III Модулі з єдиним замиканням відносно теорії скруту. III Doğruöz, S. Harmanci, A. Smith, P. F. Догруоз, С. Харманці, А. Сміт, П. Ф. We continue the study of modules over a general ring R whose submodules have a unique closure relative to a hereditary torsion theory on Mod-R. It is proved that, for a given ring R and a hereditary torsion theory τ on Mod-R, every submodule of every right R-module has a unique closure with respect to τ if and only if τ is generated by projective simple right R-modules. In particular, a ring R is a right Kasch ring if and only if every submodule of every right R-module has a unique closure with respect to the Lambek torsion theory. Продовжено вивчення модулів над загальним кільцем R, субмодулі якого мають єдине замикання відносно спадкової теорії скруту на Mod-R. Доведено, що для заданих кільця R та спадкової теорії скруту τ на Mod-R кожний субмодуль кожного правого R-модуля має єдине замикання відносно τ тоді i тільки тоді, коли τ породжується проективними простими правими R-модулями. Зокрема, кільце R є правим кільцем Каша тоді i тільки тоді, коли кожний субмодуль кожного правого R-модуля має єдине замикання відносно теорії скруту за Ламбеком. Institute of Mathematics, NAS of Ukraine 2014-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2188 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 7 (2014); 922–929 Український математичний журнал; Том 66 № 7 (2014); 922–929 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2188/1377 https://umj.imath.kiev.ua/index.php/umj/article/view/2188/1378 Copyright (c) 2014 Doğruöz S.; Harmanci A.; Smith P. F. |
| spellingShingle | Doğruöz, S. Harmanci, A. Smith, P. F. Догруоз, С. Харманці, А. Сміт, П. Ф. Modules with Unique Closure Relative to a Torsion Theory. III |
| title | Modules with Unique Closure Relative to a Torsion Theory. III |
| title_alt | Модулі з єдиним замиканням відносно теорії скруту. III |
| title_full | Modules with Unique Closure Relative to a Torsion Theory. III |
| title_fullStr | Modules with Unique Closure Relative to a Torsion Theory. III |
| title_full_unstemmed | Modules with Unique Closure Relative to a Torsion Theory. III |
| title_short | Modules with Unique Closure Relative to a Torsion Theory. III |
| title_sort | modules with unique closure relative to a torsion theory. iii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2188 |
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