“adm-n4” 22:47 page #31
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 24 (2017). Number 2, pp. 209–220
© Journal “Algebra and Discrete Mathematics”
Some remarks on Φ-sharp modules
Ahmad Yousefian Darani and Mahdi Rahmatinia
Communicated by R. Wisbauer
Abstract. The purpose of this paper is to introduce some
new classes of modules which is closely related to the classes of
sharp modules, pseudo-Dedekind modules and TV -modules. In this
paper we introduce the concepts of Φ-sharp modules, Φ-pseudo-
Dedekind modules and Φ-TV -modules. Let R be a commutative
ring with identity and set H = {M | M is an R-module and Nil(M)
is a divided prime submodule of M}. For an R-module M ∈ H,
set T = (R \ Z(M)) ∩ (R \ Z(R)), T(M) = T−1(M) and P :=
(Nil(M) :R M). In this case the mapping Φ : T(M) −→ MP given
by Φ(x/s) = x/s is an R-module homomorphism. The restriction
of Φ to M is also an R-module homomorphism from M in to MP
given by Φ(m/1) = m/1 for every m ∈ M . An R-module M ∈ H
is called a Φ-sharp module if for every nonnil submodules N,L of
M and every nonnil ideal I of R with N ⊇ IL, there exist a nonnil
ideal I ′ ⊇ I of R and a submodule L′ ⊇ L of M such that N = I ′L′.
We prove that Many of the properties and characterizations of sharp
modules may be extended to Φ-sharp modules, but some can not.
1. Introduction
We assume throughout this paper all rings are commutative with 1 6= 0
and all modules are unitary. An element x of an integral domain R is called
primal if whenever x | y1y2, with x, y1, y2 ∈ R, then x = z1z2 where z1 | y1
and z2 | y2. Cohn in [18] introduced the concept of Schreier domains.
2010 MSC: Primary 16N99, 16S99; Secondary 06C05, 16N20.
Key words and phrases: Φ-sharp module, Φ-pseudo-Dedekind module, Φ-
Dedekind module, Φ-TV module.
“adm-n4” 22:47 page #32
210 Some remarks on Φ-sharp modules
An integral domain R is called a pre-Schreier domain if every nonzero
element of R is primal. If in addition R is integrally closed, then R is
called a Schreier domain. In [27], Z. Ahmad, T. Dumitrescu and M. Epure
introduced the notion of sharp domains. A domain R is said to be a sharp
domain if whenever I ⊇ AB with I, A,B nonzero ideals of R, then there
exist ideals A′ ⊇ A and B′ ⊇ B such that I = A′B′. Let R be a ring with
identity and Nil(R) be the set of nilpotent elements of R. Recall from [19]
and [12], that a prime ideal P of R is called a divided prime ideal if P ⊂ (x)
for every x ∈ R\P ; thus a divided prime ideal is comparable to every ideal
of R. Badawi in [9], [10], [12], [13], [14] and [15], the scond-named author
investigated the class of ringsH = {R | R is a commutative ring with 1 6=
0 and Nil(R) is a divided prime ideal of R}. Anderson and Badawi in [6]
and [7] generalized the concept of Prüfer, Dedekind, Krull and Bezout
domain to context of rings that are in the class H. Lucas and Badawi
in [11] generalized the concept of Mori domains to the context of rings
that are in the class H. Also, authors this paper in [25] generalized the
concept of sharp domains to the context of rings that are in the class H.
Let R be a ring, Z(R) the set of zero divisors of R and S = R \ Z(R).
Then T (R) := S−1R denoted the total quotient ring of R. We start by
recalling some background material. A nonzero divisor of a ring R is called
a regular element and an ideal of R is said to be regular if it contains
a regular element. An ideal I of a ring R is said to be a nonnil ideal if
I * Nil(R). If I is a nonnil ideal of R ∈ H, then Nil(R) ⊂ I. In particular,
it holds if I is a regular ideal of a ring R ∈ H. Recall from [6] that for
a ring R ∈ H, the map φ : T (R) −→ RNil(R) given by φ(a/b) = a/b, for
a ∈ R and b ∈ R \Z(R), is a ring homomorphism from T (R) into RNil(R)
and φ resticted to R is also a ring homomorphism from R into RNil(R)
given by φ(x) = x/1 for every x ∈ R. Let R ∈ H. Then R is called a
φ-sharp ring if whenever for nonnil ideals I, A,B of R with I ⊇ AB, then
I = A′B′ for nonnil ideals A′, B′ of R where A′ ⊇ A and B′ ⊇ B [25].
For a nonzero ideal I of R let I−1 = {x ∈ T (R) : xI ⊆ R}. It is
obvious that II−1 ⊆ R. An ideal I of R is called invertible, if II−1 = R.
The ν-closure of I is the ideal Iν = (I−1)−1 and I is called divisorial ideal
( or ν − ideal ) if Iν = I. A nonzero ideal I of R is called t-ideal if I = It
in which
It =
⋃
{Jν | J ⊆ I is a nonzero finitely generated ideal of R}.
Let R ∈ H. Then a nonnil ideal I of R is called φ-invertible if φ(I) is an
invertible ideal of φ(R). A nonnil ideal I is φ-ν-ideal if φ(I) is a ν-ideal
of φ(R) [11]. A nonnil ideal I of R is a φ-t-ideal if φ(I) is a t-ideal of
“adm-n4” 22:47 page #33
A. Yousefian Darani, M. Rahmatinia 211
φ(R) [25]. Let R ∈ H. Then R is called a φ-pseudo-Dedekind ring if the
ν-closure of each nonnil ideal of R is φ-invertible. Also, R is said to be a
φ-TV ring in which every φ-t-ideal is a φ-ν-ideal [25].
Let R be a ring and M be an R-module. Then M is a multiplication
R-module if every submodule N of M has the form IM for some ideal
I of R. If M be a multiplication R-module and N a submodule of M ,
then N = IM for some ideal I of R. Hence I ⊆ (N :R M) and so
N = IM ⊆ (N :R M)M ⊆ N . Therefore N = (N :R M)M [16]. Let M
be a multiplication R-module, N = IM and L = JM be submodules
of M for fome ideals I and J of R. Then, the product of N and L
is denoted by N.L or NL and is defined by IJM [5]. An R-module
M is called a cancellation module if IM = JM for two ideals I and
J of R implies I = J [1]. By [21, Corollary 1 to Theorem 9], finitely
generated faithful multiplication modules are cancellation modules. It
follows that if M is a finitely generated faithful multiplication R-module,
then (IN :R M) = I(N :R M) for all ideals I of R and all submodules
N of M . If R is an integral domain and M a faithful multiplication
R-module, then M is a finitely generated R-module [17]. Let M be an
R-module and set
T = {t ∈ S : for all m ∈ M, tm = 0 implies m = 0}
= (R \ Z(M)) ∩ (R \ Z(R)).
Then T is a multiplicatively closed subset of R with T ⊆ S, and if
M is torsion-free then T = S. In particular, T = S if M is a faithful
multiplication R-module [17, Lemma 4.1]. Let N be a nonzero submodule
of M . Then we write N−1 = (M :RT N) = {x ∈ RT : xN ⊆ M}. Then
N−1 is an R-submodule of RT , R ⊆ N−1 and NN−1 ⊆ M . We say that
N is invertible in M if NN−1 = M . Clearly 0 6= M is invertible in M .
An R-module M is called a Dedekind module if every nonzero submodule
of M is invertible [20]. An R-module M is called a valuation module if
for all m,n ∈ M , either Rm ⊆ Rn or Rn ⊆ Rm. Equivalently, M is a
valuation module if for all submodules N and K of M , either N ⊆ K or
K ⊆ N [3]. The ν−closure of N is the submodule Nν = (N−1)−1 and N is
called ν−submodule if N = NνM [23] and [3]. If M is a finitely generated
faithful multiplication R-module, then Nν = (N :R M). Consequently,
Mν = R. Let M be a finitely generated faithful multiplication R-module,
N a submodule of M and I an ideal of R. Then N is a ν-submodule of
M if and only if (N :R M) is a ν-ideal of R. Also I is ν-ideal of R if and
only if IM is a ν-submodule of M [2]. If N is an invertible submodule
“adm-n4” 22:47 page #34
212 Some remarks on Φ-sharp modules
of a faithful multiplication module M over an integral domain R, then
(N :R M) is invertible and hence is a ν-ideal of R. So N is a ν-submodule
of M [2]. If R is an integral domain, M a faithful multiplication R-module
and N a nonzero submodule of M , then Nν = (N :R M)ν [2, Lemma 1].
Let M be an R-module. An element r ∈ R is said to be zero divisor
on M if rm = 0 for some 0 6= m ∈ M . The set of zero divisors of M
is denoted by ZR(M) (briefly, Z(M)). It is easy to see that Z(M) is
not necessarily an ideal of R, but it has the property that if a, b ∈ R
with ab ∈ Z(M), then either a ∈ Z(M) or b ∈ Z(M). A submodule
N of M is called a nilpotent submodule if [N :R M ]nN = 0 for some
positive integer n. An element m ∈ M is said to be nilpotent if Rm is
a nilpotent submodule of M [4]. We let Nil(M) to denote the set of all
nilpotent elements of M ; then Nil(M) is a submodule of M provided
that M is a faithful module, and if in addition M is multiplication, then
Nil(M) = Nil(R)M = ⋂P , where the intersection runs over all prime
submodules of M , [4, Theorem 6]. If M contains no nonzero nilpotent
elements, then M is called a reduced R-module. A submodule N of M is
said to be a nonnil submodule if N * Nil(M). Recall that a submodule N
of M is prime if whenever rm ∈ N for some r ∈ R and m ∈ M , then either
m ∈ N or rM ⊆ N . If N is a prime submodule of M , then p := [N :R M ]
is a prime ideal of R. In this case we say that N is a p-prime submodule
of M . Let N be a submodule of multiplication R-module M , then N is
a prime submodule of M if and only if [N :R M ] is a prime ideal of R
if and only if N = pM for some prime ideal p of R with [0 :R M ] ⊆ p,
[17, Corollary 2.11]. Recall from [3] that a prime submodule P of M is
called a divided prime submodule if P ⊂ Rm for every m ∈ M \ P ; thus
a divided prime submodule is comparable to every submodule of M .
Now assume that T−1(M) = T(M). Set
H = {M | M is an R-module and Nil(M) is a divided prime
submodule of M}.
For an R-module M ∈ H, Nil(M) is a prime submodule of M . So P :=
[Nil(M) :R M ] is a prime ideal of R. If M is an R-module and Nil(M) is
a proper submodule of M , then [Nil(M) :R M ] ⊆ Z(R). Consequently,
R \ Z(R) ⊆ R \ [Nil(M) :R M ]. In particular, T ⊆ R \ [Nil(M) :R M ]
[22]. Recall from [22] that we can define a mapping Φ : T(M) −→ MP
given by Φ(x/s) = x/s which is clearly an R-module homomorphism.
The restriction of Φ to M is also an R-module homomorphism from M
in to MP given by Φ(m/1) = m/1 for every m ∈ M . A nonnil submodule
“adm-n4” 22:47 page #35
A. Yousefian Darani, M. Rahmatinia 213
N of M is said to be Φ-invertible if Φ(N) is an invertible submodule
of Φ(M) [26]. Let M ∈ H. Then M is a Φ-Dedekind R-module if every
nonnil submodule of M is Φ-invertible [26]. In this paper we introduce a
generalization of φ-sharp rings and give some properties of this class of
modules.
2. Φ-sharp modules
Definition 2.1. Let R be a ring and M ∈ H be an R-module. Then M
is called a Φ-sharp module if for every nonnil submodules N,L of M and
every nonnil ideal I of R with N ⊇ IL, there exist a nonnil ideal I ′ ⊇ I
of R and a submodule L′ ⊇ L of M such that N = I ′L′.
Theorem 2.2. Let R be a ring and M ∈ H with Nil(M) = Z(R)M .
Then M is a Φ-sharp module if and only if M/Nil(M) is a sharp module.
Proof. Since Nil(M) = Z(R)M , then Nil(R) = (Nil(M) :R M) =
(Z(R)M :R M) = Z(R) by [22, Proposition 1]. LetM be a Φ-sharp module
and let N/Nil(M), L/Nil(M) be nonzero submodules of M/Nil(M) and
I be a nonzero ideal of R with N/Nil(M) ⊇ I(L/Nil(M)). Then N ⊇ IL
and so there exist a nonnil ideal I ′ ⊇ I of R and a submodule L′ ⊇ L of
M such that N = I ′L′. Thus N/Nil(M) = I ′((L′/Nil(M)) for nonzero
ideal I ′ ⊇ I of R and for a nonzero submodule L/Nil(M) ⊇ L′/Nil(M)
of M/Nil(M) as well.
Conversely, let M/Nil(M) be a sharp module and let N,L be nonnil
submodules of M and I a nonnil ideal of R such that N ⊇ IL. Then
N/Nil(M), L/Nil(M) are nonzero submodules of M/Nil(M) and I is a
nonzero ideal of R with N/Nil(M) ⊇ I(L/Nil(M)). So, N/Nil(M) =
I ′((L′/Nil(M)) for nonzero ideal I ′ ⊇ I of R and for a nonzero submodule
L/Nil(M) ⊇ L′/Nil(M) of M/Nil(M). Therefore N = I ′L′ for a nonnil
ideal I ′ ⊇ I of R and for a submodule L′ ⊇ L of M . Thus M is a Φ-sharp
module.
Lemma 2.3. ([26, Lemma 2.6]) Let R be a ring and M a finitely gen-
erated faithful multiplication R-module with M ∈ H. Then MNil(M) is
isomorphic to Φ(M)Nil(Φ(M)) as R-module.
Corollary 2.4. Let R be a ring and M ∈ H be a finitely generated faithful
multiplication R-module with Nil(M) = Z(R)M . Then M is a Φ-sharp
module if and only if Φ(M)Nil(Φ(M)) is a sharp module.
“adm-n4” 22:47 page #36
214 Some remarks on Φ-sharp modules
Theorem 2.5. Let R be a ring and M ∈ H with Nil(M) = Z(R)M .
Then M is a Φ-sharp module if and only if Φ(M) is a sharp module.
Proof. Let M be a Φ-sharp module and let Φ(N) ⊇ IΦ(L) for nonnil
submodules N,L of M and nonnil ideal I of R. Since Nil(M) is a divided
prime submodule of M and N,L properly contain Nil(M), so both contain
Ker(Φ) by [26, Propoition 2.1]. Therefore N ⊇ IL and hence N = I ′L′
for a nonnil submodule L′ ⊇ L of M and a nonnil ideal I ′ ⊇ I of R. Thus
Φ(N) = I ′Φ(L′) for a submodule Φ(L′) ⊇ Φ(L) and an ideal I ′ ⊇ I. So
Φ(M) is a sharp module.
Converesly, Let Φ(M) be a sharp module and let N,L be nonnil
submodules of M and I an ideal of R with N ⊇ IL. Thus Φ(N) ⊇ IΦ(L)
and so Φ(N) = I ′Φ(L′) for a submodule Φ(L′) ⊇ Φ(L) and an ideal
I ′ ⊇ I. By the same reason as above, we have N = I ′L′ for a nonnil
submodule L′ ⊇ L of M and a nonnil ideal I ′ ⊇ I of R. Hence M is a
Φ-sharp module.
Corollary 2.6. Let R be a ring and M ∈ H be a finitely generated
faithful multiplication R-module with Nil(M) = Z(R)M . The following
statements are equivalent:
(1) M is a Φ-sharp module;
(2) M/Nil(M) is a sharp module;
(3) Φ(M)Nil(Φ(M)) is a sharp module;
(4) Φ(M) is a sharp module.
Proposition 2.7. Let R be a ring and M ∈ H be a finitely generated
faithful multiplication R-module with Nil(M) = Z(R)M . If M is a Φ-
Dedekind module, then M is a Φ-sharp module.
Proof. If M is a Φ-Dedekind module, then M/Nil(M) is a Dedekind
module by [26, Theorem 2.10]. So, by [23, Corollary 3.5], M/Nil(M) is a
sharp module. Therefore, by Theorem 2.2, M is a Φ-sharp module.
In [26] it is shown that for each prime ideal P of R, (M/Nil(M))P =
MP /(Nil(M))P = MP /Nil(MP ) and MP ∈ H.
Proposition 2.8. Let R be a ring and M ∈ H be a Φ-sharp module with
Nil(M) = Z(R)M . Then MP is a Φ-sharp module for each prime ideal
P of R.
Proof. We have Nil(R) ⊆ Ann( MNil(R)M ) = Ann( MNil(M)). If M is a Φ-
sharp module, then by Theorem 2.2, M/Nil(M) is a sharp module. So, by
“adm-n4” 22:47 page #37
A. Yousefian Darani, M. Rahmatinia 215
[23, Proposition 3.8], (M/Nil(M))P = MP /Nil(MP ) is a sharp module.
Therefore, by Theorem 2.2, MP is a Φ-sharp module.
Theorem 2.9. Let R be a ring and M be a finitely generated faithful
multiplication R-module. The following statements are equivalent:
(1) If R ∈ H is a φ-sharp ring, then M is a Φ-sharp module;
(2) If M ∈ H is a Φ-sharp module, then R is a φ-sharp ring.
Proof. (1) ⇒ (2) Let R ∈ H. Then, by [22, Proposition 3], M ∈ H. Let
R be a φ-sharp ring and let N,L be nonnil submodules of M and I be
a nonnil ideal of R with N ⊇ IL. Then (N :R M), (L :R M) are nonnil
ideals of R such that (N :R M) ⊇ I(L :R M). So (N :R M) = I ′J ′
for nonnil ideals I ′ ⊇ I and J ′ ⊇ (L :R M) of R. Thus N = I ′(J ′M)
for a nonnil ideal I ′ ⊇ I of R and a nonnil submodule J ′M ⊇ L of M .
Therefore M is a Φ-sharp module.
(2) ⇒ (1) Let M ∈ H. Then, by [22, Proposition 3], R ∈ H. Let M be
a Φ-sharp module and let I, J,K be nonnil ideals of R with K ⊇ IJ . So
KM,JM are nonnil submodules of M such that KM ⊇ I(JM). Thus
KM = I ′L′ for a nonnil ideal I ′ ⊇ I of R and a nonnil submodule
L′ ⊇ JM of M . Therefore K = I ′(L′ :R M) for nonnil ideals I ′ ⊇ I and
(L′ :R M) ⊇ J of R. So R is a φ-sharp ring.
Definition 2.10. Let R be a ring and M be an R-module. Then M is
said to be a Φ-pseudo-Dedekind module if the ν-closure of each nonnil
submodule of M is Φ-invertible.
Theorem 2.11. Let R be a ring and M ∈ H be an R-module. Then M
is a Φ-pseudo-Dedekind module if and only if M/Nil(M) is a pseudo-
Dedekind module.
Proof. Let M be a Φ-pseudo-Dedekind module and N/Nil(M) be a
nonzero submodule of M/Nil(M). Then N is a nonnil submodule of M
and so the ν-closure of N is Φ-invertible, i.e, Nν is Φ-invertible. Thus, by
[24, Lemma 3.6], (N/Nil(M))ν = Nν/Nil(M) is invertible as well.
Conversely, let M/Nil(M) be a pseudo-Dedekind module and N be
a nonnil submodule of M . Thus N/Nil(M) is a nonzero submodule
of M/Nil(M) and so Nν/Nil(M) = (N/Nil(M))ν is invertible. So, by
[24, Lemma 3.6], Nν is Φ-invertible. Therefore, M is a Φ-pseudo-Dedekind
module.
By Lemma 2.3, we have the following theorem.
“adm-n4” 22:47 page #38
216 Some remarks on Φ-sharp modules
Corollary 2.12. Let R be a ring and M ∈ H be a finitely generated
faithful multiplication R-module. Then M is a Φ-pseudo-Dedekind module
if and only if Φ(M)Nil(Φ(M)) is a pseudo-Dedekind module.
Theorem 2.13. Let R be a ring and M ∈ H be an R-module. Then M
is a Φ-pseudo-Dedekind module if and only if Φ(M) is a pseudo-Dedekind
module.
Proof. Let M be a Φ-pseudo-Dedekind module and Φ(N) be a submodule
of Φ(M) for a nonnil submodule N of M . Thus Nν is Φ-invertible. Hence
Φ(Nν) = (Φ(N))ν is invertible.
Conversely, let Φ(M) be a pseudo-Dedekind module and N be a
nonnil submodule of M . Then Φ(N) is a submodule of Φ(M) and so
(Φ(N))ν = Φ(Nν) is invertible submodule of Φ(M). Therefore Nν is
Φ-invertible.
Corollary 2.14. Let R be a ring and M ∈ H be a finitely generated
faithful multiplication R-module. The following are equivalent:
(1) M is a Φ-pseudo-Dedekind module;
(2) M/Nil(M) is a pseudo-Dedekind module;
(3) Φ(M)/Nil(Φ(M)) is a pseudo-Dedekind module;
(4) Φ(M) is a pseudo-Dedekind module.
Theorem 2.15. Let R be a ring and M be a finitely generated faithful
multiplication R-module. The following statements are equivalent:
(1) If R ∈ H is a φ-pseudo-Dedekind ring, then M is a Φ-pseudo-
Dedekind module;
(2) If M ∈ H is a Φ-pseudo-Dedekind module, then R is a φ-pseudo-
Dedekind ring.
Proof. Since Nil(R) ⊆ Ann( MNil(R)M ) = Ann( MNil(M)), we have:
(1) ⇒ (2) Let R ∈ H. Then, by [22, Proposition 3], M ∈ H. If R is a
φ-pseudo-Dedekind ring, then by [25, Theorem 2.10], RNil(R) is a pseudo-
Dedekind domain. So, by [23, Theorem 3.12], MNil(M) is a pseudo-Dedekind
module. Therefore, by Theorem 2.11, M is a Φ-pseudo-Dedekind module.
(2) ⇒ (1) Let M ∈ H. Then, by [22, Proposition 3], R ∈ H. If M is a
Φ-pseudo-Dedekind module, then by Theorem 2.11, MNil(M) is a pseudo-
Dedekind module. So, by [23, Theorem 3.12], RNil(R) is a pseudo-Dedekind
domain. Therefore, by [25, Theorem 2.10], R is a φ-pseudo-Dedekind
ring.
“adm-n4” 22:47 page #39
A. Yousefian Darani, M. Rahmatinia 217
Proposition 2.16. Let R be a ring and M ∈ H be a finitely generated
faithful multiplication R-module. If M is a Φ-sharp module, then M is a
Φ-pseudo-Dedekind module.
Proof. Let M be a Φ-sharp module. Then, by Theorem 2.2, M/Nil(M)
is a sharp module. So, by [23, Lemma 3.11], M/Nil(M) is a pseudo-
Dedekind module. Therefore, by Theorem 2.11,M is a Φ-pseudo-Dedekind
module.
Recall from [26], an R-module M ∈ H is called a Φ-valuation module if
for every u ∈ R(Nil(R):RM), we have uΦ(M) ⊆ Φ(M) or u−1Φ(M) ⊆ Φ(M);
equivalently, for every a, b /∈ (Nil(R) :R M), either, aΦ(M) ⊆ bΦ(M) or
bΦ(M) ⊆ aΦ(M).
Theorem 2.17. Let R be a ring and M ∈ H be a finitely generated faithful
multiplication Φ-valuation R-module. Then the following are equivalent:
(1) M is a Φ-sharp module;
(2) M is a Φ-pseudo-Dedekind module.
Proof. (1) ⇒ (2) is given by Proposition 2.16.
(2) ⇒ (1) Let M is a Φ-pseudo-Dedekind module. Then, by The-
orem 2.11, M/Nil(M) is a pseudo-Dedekind-module. Since M is a Φ-
valuation module, then by [26, Theorem 2.13], M/Nil(M) is a Valuation
module. So M/Nil(M) is sharp module by [23, Proposition 3.14]. There-
fore, by Theorem 2.11, M is a Φ-sharp module.
Definition 2.18. Let R be a ring and M ∈ H be an R-module. A
nonnil submodule N of M is called a Φ-t-submodule of M if Φ(N) is a
t-submodule of Φ(M).
It is worthwhile to note that N/Nil(M) is a t-submodule of M/Nil(M)
if and only if Φ(N)/Nil(Φ(M)) is a t-submodule of Φ(M)/Nil(Φ(M)).
Lemma 2.19. Let R be a ring and M ∈ H be an R-module and let N be
a nonnil submodule of M . Then N is a Φ-t-submodule of M if and only
if N/Nil(M) is a t-submodule of M/Nil(M).
Proof. Let N be a Φ-t-submodule of M . Then Φ(N) is a t-submodule of
Φ(M). Thus Φ(N) = Φ(N)νΦ(M) and so
Φ(N)/Nil(Φ(M)) = (Φ(N)ν/Nil(Φ(M)))(Φ(M)/Nil(Φ(M))).
Therefore Φ(N)/Nil(Φ(M)) is a t-submodule of Φ(M)/Nil(Φ(M)). Hence
N/Nil(M) is a t-submodule of M/Nil(M). Conversely is same.
“adm-n4” 22:47 page #40
218 Some remarks on Φ-sharp modules
Definition 2.20. Let R be a ring and M ∈ H be an R-module. Then M
is said to be a Φ-TV module if every Φ-t-submodule is a Φ-ν-submodule.
Theorem 2.21. Let R be a ring and M ∈ H be an R-module. Then M
is a Φ-TV module if and only if M/Nil(M) is a TV -module.
Proof. Let M be a Φ-TV module and N/Nil(M) be a t-submodule of
M/Nil(M). Then, by Lemma 2.19, N a is Φ-t-submodule of M and so
N is a Φ-ν-submodule of M . Hence, by [24, Lemma 3.6], N/Nil(M) is a
ν-submodule of M/Nil(M). Thus M/Nil(M) is a TV -module.
Conversely, letM/Nil(M) be a TV -module andN be a Φ-t-submodule
of M . Then, by Lemma 2.19, N/Nil(M) is a t-submodule of M/Nil(M)
and so N/Nil(M) is a ν-submodule of M/Nil(M). Therefore, by [24,
Lemma 3.6], N is a Φ-t-submodule of M as well.
Corollary 2.22. Let R be a ring and M ∈ H be an R-module. Then M
is a Φ-TV module if and only if Φ(M)/Nil(Φ(M)) is a TV -module.
Theorem 2.23. Let R be a ring and M ∈ H be an R-module. Then M
is a Φ-TV module if and only if Φ(M) is a TV module.
Proof. Let M be a Φ-TV module and Φ(N) be a t-submodule of Φ(M).
Then N is a Φ-t-submodule of M and so N is a Φ-ν-submodule of M .
Therefore, Φ(N) is a ν-submodule of Φ(M). Hence Φ(M) is a TV module.
Conversely, let Φ(M) be a TV module and N be a Φ-t-submodule of
M . Then Φ(N) is a t-submodule of Φ(M) and so Φ(N) is a ν-submodule
of Φ(M). Thus N is a Φ-ν-submodule of M . Therefore M is a Φ-TV
module.
Corollary 2.24. Let R be a ring and M ∈ H be a finitely generated
faithful multiplication R-module. The following are equivalent:
(1) M is a Φ-TV module;
(2) M/Nil(M) is a TV module;
(3) Φ(M)/Nil(Φ(M)) is a TV module;
(4) Φ(M) is a TV module.
Theorem 2.25. Let R be a ring and M be a finitely generated faithful
multiplication R-module. The following statements are equivalent:
(1) If R ∈ H is a φ-TV ring, then M is a Φ-TV module;
(2) If M ∈ H is a Φ-TV module, then R is a φ-TV ring.
Proof. By [22], [23] and [25], the proof is the same of the proof of Theo-
rem 2.15.
“adm-n4” 22:47 page #41
A. Yousefian Darani, M. Rahmatinia 219
The notion of a Φ-sharp-TV module means that a module that is
both a Φ-sharp module and a Φ-TV module.
Theorem 2.26. Let R be a ring and M ∈ H be a finitely generated faithful
multiplication R-module with Nil(M) = Z(R)M . If M is a Φ-sharp TV
module, then M is a Φ-Dedekind module.
Proof. Let M be a Φ-sharp TV module. Then, by Theorem 2.2 and The-
orem 2.21, M/Nil(M) is a sharp TV module. So, by [23, Corollary 3.21],
M/Nil(M) is a Dedekind module. Therefore M is a Φ-Dedekind module
by [26, Theorem2.10].
Theorem 2.27. Let R be a countable ring and M ∈ H be an R-module
with Nil(M) = Z(R)M . If M is a Φ-sharp module, then M is a Φ-
Dedekind module.
Proof. If M is a Φ-sharp module, then M/Nil(M) is a sharp module by
Theorem 2.2. So, by [23, Theorem 3.7], R is a sharp domain and hence
by [27, Corollary 17], R is a Dedekind domain. Thus M/Nil(M) is a
Dedekind domain. Therefore, by [26, Theorem2.10], M is a Φ-Dedekind
module.
References
[1] M. M. Ali, Some remarks on generalized GCD domains, Comm. Algebra, 36
(2008), 142–164.
[2] M. M. Ali, Invertibility of multiplication modules Π, New Zealand J. Math., 39
(2009), 45–64.
[3] M. M. Ali, Invertibility of multiplication modules III, New Zealand J. Math., 39
(2009), 139–213.
[4] M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm.
Algebra, 36 (2008), 4620–4642.
[5] R. Ameri, On the prime submodules of multiplication modules, IJMMS, 27 (2003),
1715–1724.
[6] D. F. Anderson and A. Badawi, On φ-Prüfer rings and φ-Bezout rings, Houston J.
math. 2 (2004), 331–343.
[7] D. F. Anderson and A. Badawi, On φ-Dedekind rings and φ-Krull rings, Houston
J. math. 4 (2005), 1007–1022.
[8] D. F. Anderson and V. Barucci and D. D. Dobbs, Coherent Mori domain and the
principal ideal theorem, Comm. Algebra 15 (1987), 1119–1156.
[9] A. Badawi, On φ-pseudo- valuation rings, Lecture Notes Pure Appl. Math., vol
205 (1999), 101–110, Marcel Dekker, New York/Basel.
[10] A. Badawi, On φ-pseudo- valuation rings II, Houston J. Math. 26 (2000), 473–480.
“adm-n4” 22:47 page #42
220 Some remarks on Φ-sharp modules
[11] A. Badawi and Thomas G. Lucas, On φ-Mori rings, Houston J. math. 32 (2006),
1–32.
[12] A. Badawi, On divided commutative rings, Comm. Algebra, 27 (1999), 1465–1474.
[13] A. Badawi, On φ-chained rings and φ-pseudo-valuation rings, Houston J. math.
27 (2001), 725–736.
[14] A. Badawi, On divided rings and φ-pseudo-valuation rings, International J of
Commutative Rings(IJCR), 1 (2002), 51–60.
[15] A. Badawi, On nonnil-Noetherian rings, Comm. Algebra, 31 (2003), 1669–1677.
[16] A. Barnard, Multiplication modules, J. Algebra, 71 (1981), 174-178.
[17] Z. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16 (1998),
755–799.
[18] P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc, 64
(1968), 251-264.
[19] D. E. Dobbs, Divided rings and going-down, Pacific J. math. 67 (1976), 353–363.
[20] A. G. Naoum and F. H. Al-Alwan, Dedekind modules, Comm. Algebra, 24 (1996),
225–230.
[21] P. F. Smith, Some remarks on multiplication modules, Arch. der. Math., 50 (1988),
223-235.
[22] A. Youseffian Darani, Nonnil-Noetherian modules over commutative rings, Sub-
mitted.
[23] A. Youseffian Darani and M. Rahmatinia, On sharp modules over commutative
rings, Submitted.
[24] A. Youseffian Darani and M. Rahmatinia, On Φ-Mori modules, Submitted.
[25] A. Youseffian Darani and M. Rahmatinia, On Φ-sharp rings, Submitted.
[26] A. Youseffian Darani and S. Motmaen, On Φ-Dedekind, φ-Prüfer and Φ-Bezout
modules , Submitted.
[27] A. Zaheer, D. Teberiu and E. Mihai, A schreier domain type condition, Bull. Math.
Soc. Roumania, 55(3) (2012), 241–247.
Contact information
A. Yousefian
Darani,
M. Rahmatinia
Department of Mathematics and Applications,
University of Mohaghegh Ardabili, P. O. Box
179, Ardabil, Iran
E-Mail(s): yousefian@uma.ac.ir,
m.rahmati@uma.ac.ir
Received by the editors: 27.11.2015.