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Mathematical Publications
DOI: 10.2478/tmmp-2024-0023
Tatra Mt. Math. Publ. 87 (2024), 85–104
GRADED CLASSICAL WEAKLY PRIME
SUBMODULES
OVER NON-COMMUTATIVE GRADED RINGS
Jebrel M. Habeb — Rashid Abu-Dawwas
Department of Mathematics, Yarmouk University, Irbid, JORDAN
This work had been done while the first author had a sabbatical leave from Yarmouk
University for the academic year 2022-2023.
ABSTRACT. The goal of this article is to propose and examine the notion
of graded classical weakly prime submodules over non-commutative graded rings
which is a generalization of the concept of graded classical weakly prime submod-
ules over commutative graded rings. We investigate the structure of these types
of submodules in various categories of graded modules.
1. Introduction
Throughout this article, all rings considered are associative and have a nonzero
unity. Similarly, all modules are unital left modules. Let Gbe a multiplicative
group with identity e,andletAbe a ring with nonzero unity 1. A ring Ais called
G-graded if
A=
g∈G
Agwith AgAh⊆Agh for all g, h ∈G,
where Agis an additive subgroup of Afor each g∈G.HereAgAhdenotes the set
of all finite sums of elements agbh,whereag∈Agand bh∈Ah. This structure
is denoted as G(A). The elements of Agare called homogeneous of degree g.
If a∈A, it can be uniquely written as a=g∈Gag,whereagis the component
©
2024 Mathematical Institute, Slovak Academy of Sciences.
2020 M a t h e m a t i c s Subject Classification: 16W50, 13A02, 13A15.
K e yw o r d s: graded prime submodules, graded weakly prime submodules, graded classical
prime submodules.
Licensed under the Creati ve Commons BY-NC-ND 4.0 International Public License.
85
J.M. HABEB—R. ABU-DAWWAS
of ain Ag,andag= 0 for all but finitely many g. The additive subgroup Aeis a
subring of Aand 1 ∈Ae. The set of all homogeneous elements of Ais g∈GAg
and is denoted by h(A). Let Pbe a left ideal of a G-graded ring A.ThenPis
called a graded left ideal if
P=
g∈G
(P∩Ag),i.e., for a∈P, a =
g∈G
ag,
where ag∈Pfor all g∈G. Not all left ideals of a graded ring are necessarily
graded left ideals (see Example 1.1 in [3]).
Let Abe a G-graded ring. A left A-module Mis called G-graded if
M=
g∈G
Mgwith AgMh⊆Mgh; for all g, h ∈G,
where Mgis an additive subgroup of Mfor all g∈G. The elements of Mgare
called homogeneous of degree g.Ifm∈M,thenmcan be written uniquely as
m=g∈Gmg,wheremgis the component of min Mgand mg= 0 for all but
finitely many g.EachMgis an Ae-module. The set of all homogeneous elements
of Mis g∈GMgand is denoted by h(M). Let Kbe an A-submodule of M.
Kis called a graded submodule of Mif
K=
g∈G
(K∩Mg),i.e., for x∈K, x =
g∈G
xg
where xg∈Kfor all g∈G. As with graded ideals, not all A-submodules
of a graded A-module are necessarily graded submodules. For further details
and terminology, see [10, 14].
The concept of graded prime ideals in commutative graded rings was in-
troduced in [17]. A proper graded ideal Pof a commutative graded ring A
is said to be graded prime if, whenever x, y ∈h(A) such that xy ∈P,then
either x∈Por y∈P. This notion and its generalizations are central in graded
commutative algebra, as they provide valuable tools for studying the proper-
ties of graded commutative rings. Several generalizations of graded prime ideals
have been proposed and explored. For instance, Atani introduced the concept
of graded weakly prime ideals in [5]. A proper graded ideal Pof a commutative
graded ring Ais said to be a graded weakly prime ideal if, whenever x, y ∈h(A)
and 0 =xy ∈P,thenx∈Por y∈P. By ([5], Theorem 2.12), the following
statements are equivalent for a graded ideal Pof G(A)withP=A,whereAis
a commutative graded ring:
(1) Pis a graded weakly prime ideal of G(A).
(2) For each g, h ∈G, the inclusion 0 =IJ ⊆Pwith Ae-submodules Iof Ag
and Jof Ahimplies that I⊆Por J⊆P.
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GRADED CLASSICAL WEAKLY PRIME SUBMODULES
For graded rings that are not necessarily commutative, it is clear that (2)
does not imply (1). In [3], Alshehry and Abu-Dawwas defined a graded weakly
prime ideal for non-commutative graded rings as follows: a graded left ideal P
of G(A)withP=Ais said to be a graded weakly prime ideal of G(A) if for
each g, h ∈G, the inclusion 0 =IJ ⊆Pwith Ae-submodules Iof Agand J
of Ahimplies that I⊆Por J⊆P. Equivalently, whenever x, y ∈h(A)such
that 0 =xAy ⊆P, then either x∈Por y∈P([3, Proposition 2.3]). In [2],
the standard definition of a graded prime ideal Pfor a graded non-commutative
ring Ais that P=Aand whenever Iand Jare graded left ideals of Asuch
that IJ ⊆P, then either I⊆Por J⊆P. Accordingly, in [3], they defined
a graded left ideal of a graded ring Ato be a graded weakly prime as follows:
a proper graded left ideal Pof Ais said to be a graded weakly prime ideal of
Aif whenever Iand Jare graded left ideals of Asuch that 0 =IJ ⊆P,then
either I⊆Por J⊆P.
Let Kbe a graded A-submodule of a left A-module M.Then(K:AM)=
{a∈A:aM ⊆K}is a graded two-sided ideal of A[8]. Kis said to be faithful
if AnnA(K)=(0:
AK) = 0. In [2], a proper graded A-submodule Kof a graded
A-module Mover a non-commutative graded ring Ais said to be graded prime
if whenever Lis a graded A-submodule of Mand Iis a graded ideal of Asuch
that IL ⊆K, then either L⊆Kor I⊆(K:AM). If Ais commutative, this
definition is equivalent to: a proper graded A-submodule Kof a graded A-module
Mis said to be graded prime if whenever a∈h(A)andx∈h(M)aresuchthat
ax ∈K, then either x∈Kor a∈(K:AM). In a similar way, a proper graded
A-submodule Kof a graded A-module Mover a non-commutative graded ring A
is said to be graded weakly prime if whenever Lis a graded A-submodule of M
and Iis a graded ideal of Asuch that 0 =IL ⊆K, then either L⊆Kor
I⊆(K:AM). If Ais commutative, this definition is equivalent to: a proper
graded A-submodule KofagradedA-module Mis said to be graded weakly
prime if whenever a∈h(A)andx∈h(M)aresuchthat0=ax ∈K, then either
x∈Kor a∈(K:AM). Let Kbe a graded A-submodule of Mand g∈Gsuch
that Kg=Mg.ThenKis said to be a g-prime A-submodule of Mif whenever
Lis an Ae-submodule of Mgand Iis an ideal of Aesuch that IL ⊆K,then
either L⊆Kor I⊆(K:AM).
The concept of graded weakly classical prime submodules over commutative
graded rings has been proposed and studied by Abu-Dawwas and Al-Zoubi in [1].
A proper graded A-submodule Kof Mover a commutative graded ring Ais said
to be a graded weakly classical prime if whenever x, y ∈h(A)andm∈h(M)
such that 0 =xym ∈K, then either xm ∈Kor ym ∈K. In this article, we intro-
duce and examine the concept of graded classical weakly prime submodules over
non-commutative graded rings. Indeed, this article is motivated by the concepts
and the techniques that have been examined in [11]. We propose the following:
87
J.M. HABEB—R. ABU-DAWWAS
a proper graded A-submodule Kof Mover a non-commutative graded ring A
is said to be a graded classical weakly prime if whenever x, y ∈h(A)andLis
agradedA-submodule of Msuch that 0 =xAyL ⊆K, then either xL ⊆Kor
yL ⊆K. Several properties have been examined. Also, we investigate the struc-
ture of graded classical weakly prime submodules in various categories of graded
modules.
2. Graded classical weakly prime submodules
In this section, we introduce and examine graded classical weakly prime sub-
modules over non-commutative graded rings.
2.1
Let Abe a graded ring, Mbe a graded A-module, Kbe
agradedA-submodule of M,andg∈G.Then
(1) Kis called a graded classical weakly prime A-submodule of Mif K=M
and whenever x, y ∈h(A)andLis a graded A-submodule of Msuch that
0=xAyL ⊆K, then either xL ⊆Kor yL ⊆K.
(2) Kis called a graded completely classical weakly prime A-submodule of M
if K=Mand whenever x, y ∈h(A)andz∈h(M) such that 0 =xyz ∈K,
then either xz ∈Kor yz ∈K.
(3) Kis called a g-classical weakly prime A-submodule of Mif Kg=Mg
and whenever x, y ∈Aeand Lis an Ae-submodule of Mgsuch that
0=xAeyL ⊆K, then either xL ⊆Kor yL ⊆K.
Clearly, every graded prime A-submodule of Mis a graded classical weakly
prime A-submodule of M. Additionally, if Ais a commutative graded ring with
unity, then the concepts of graded classical weakly prime submodules and graded
completely classical weakly prime submodules coincide. However, the follow-
ing two examples demonstrate that this equivalence does not hold for non-
commutative graded rings.
Example 2.2.Consider A=M=M2(Z)(The2×2 matrices over the ring
of integers Z)andG=Z4(The additive group of integers modulo 4). The ring
Ais G-graded as follows:
A0=Z0
0Z,A
2=0Z
Z0and A1=A3=00
00
.
The module Mis also G-graded with the same grading as A.LetK=M2(2Z).
Then Kis a graded prime A-submodule of M, and hence Kis a graded clas-
sical weakly prime A-submodule of M. On the other hand, Kis not a graded
88
GRADED CLASSICAL WEAKLY PRIME SUBMODULES
completely classical weakly prime A-submodule of Msince
x=10
02
∈h(A),y=01
10
∈h(A)andz=10
04
∈h(M)
are such that 0 =xyz ∈K,xz /∈Kand yz /∈K.
Example 2.3.Let A=M=M3(Z)andG=Z. The ring Ais G-graded as
follows:
A0=⎛
⎝
Z00
0Z0
00Z
⎞
⎠,A
1=⎛
⎝
0Z0
00Z
000
⎞
⎠,
A2=⎛
⎝
00Z
000
000
⎞
⎠,A
n!=⎛
⎝
000
000
000
⎞
⎠,
for |n|>2andA−n=At
n(the transpose of An), for n=1,2. Then K=M3(2Z)
is a graded prime A-submodule of M, and thus Kis a graded classical weakly
prime A-submodule of M. On the other hand, Kis not a graded completely
classical weakly prime A-submodule of Msince
x=⎛
⎝
100
020
001
⎞
⎠∈h(A),y=⎛
⎝
010
001
000
⎞
⎠∈h(A)
and
z=⎛
⎝
000
000
300
⎞
⎠∈h(M)
are such that 0 =xyz ∈K,xz /∈Kand yz /∈K.
Example 2.4.Consider the ring A=M3(F) (the ring of 3 ×3 matrices over
afieldF) and the group G=Z. To construct a grading of Aby Gwhere all
components are nonzero, we proceed as follows:
(1) A standard basis for M3(F) is given by the elementary matrices eij ,where
eij has 1 in the (i, j)-th position and 0 elsewhere. The set {eij |1≤i, j ≤3}
forms a basis for the vector space M3(F).
(2) We assign degrees from Zto each matrix entry in such a way that no
component is zero. Specifically, we define the grading as:
A=
g∈Z
Ag,
where Ag=span{eij |deg(eij )=g}. The degree of each elementary
matrix eij is given by a degree map deg : {eij }→Z.
89
J.M. HABEB—R. ABU-DAWWAS
(3) Using deg(eij)=j−i, we assign the following degrees to the elementary
matrices:
deg(e11)= 0,deg(e12)= 1,deg(e13)=2,
deg(e21)=−1,deg(e22)= 0,deg(e23)=1,
deg(e31)=−2,deg(e32)=−1,deg(e33)=0.
(4) With the degree assignment above, the graded components Agfor g∈Zare
A0=span{e11,e
22,e
33},A
1=span{e12,e
23},A
2=span{e13},
A−1=span{e21,e
32},A
−2=span{e31},
and other components are determined by the property
AgAh=Ag+h.
All graded components are nonzero, which satisfies the requirement.
2.5
[2] Let Abe a graded ring, Mbe a graded A-module, Kbe
agradedA-submodule of Mand g∈G.ThenKis said to be a g-prime
A-submodule of Mif Kg=Mgand whenever Iis an ideal of Aeand Lis
an Ae-submodule of Mgsuch that IL⊆K, then either
L⊆Kor I⊆(K:AM)= {a∈A:aM ⊆K}.
Clearly, the zero submodule is always graded classical weakly prime and
g-classical weakly prime for all g∈Gby definition. However, the following
example demonstrates that one can find g∈Gsuch that the zero submodule is
not a g-prime.
Example 2.6.Consider A=M=M2(Z)andG=Z4.ThenAis G-graded by
A0=Z0
0Z,A
2=0Z
Z0and A1=A3=00
00
.
Malso is a G-graded left A-module by the same graduation of A.Choose
x=00
01
∈A0.
Then
I=A0xA0=00
0Zis an ideal of A0.
Choose
y=01
00
∈M2.
Then
L=A0y=0Z
00
is an A0-submodule of M2.
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GRADED CLASSICAL WEAKLY PRIME SUBMODULES
Consider the graded A-submodule
K=00
00
of M.
Then
K2=00
00
=M2and IL ⊆K,
but LKand I(K:AM). Hence, Kis not a 2-prime A-submodule of M.
2.7
Let Mbe a graded A-module, Kbe a graded A-submodule of M
and g∈Gsuch that Kg=Mgand every Ae-submodule of Mgis faithful.
Then Kis a g-classica l weakly pri me A-submodule of Mif and only if, whenever
I,J are ideals of Aeand Lis an Ae-submodule of Mgwith 0=IJL ⊆K,
we have either IL ⊆Kor JL ⊆K.
P r o o f. Suppose Kis a g-classical weakly prime A-submodule of M.LetI, J
be ideals of Aeand Lan Ae-submodule of Mgwith 0 =IJL ⊆K. Suppose
that IL Kand JL K. Then there exist r∈Iand s∈Jsuch that
rL Kand sL K.SincerAesL ⊆IJL ⊆K,andKis g-weakly classical
prime, this implies rAesL =0,sorsL =0.Thus,rs =0.ToshowIJ =0,
let a∈Iand b∈J.IfaL Kand bL K,thenab = 0 as shown earlier.
If aL Kand bL ⊆K,then(s+b)LK,andthena(s+b) = 0. Since
as =0,ab = 0. Similarly, if aL ⊆Kand bL K,thenab =0.IfaL ⊆K
and bL ⊆K,then(r+a)LKand (s+b)LK,andthen(r+a)s=
r(s+b)=(r+a)(s+b)=rs = 0, which gives that ab =0.ThusIJ =0,
and then IJL = 0, which is a contradiction. Hence, either IL ⊆Kor JL ⊆K.
Conversely, let x, y ∈Aeand Lbe an Ae-submodule Mgwith 0 =xAeyL ⊆K.
Then I=Aexand J=Aeyare ideals of Aewith 0 =IJL ⊆K,andthen
either IL ⊆Kor JL ⊆Kby assumption, and hence either xL ⊆Kor yL ⊆K.
Thus Kis a g-classical weakly prime A-submodule of M.
Remark 2.8
Upon careful examination, we could not identify an example
where the faithfulness condition in Theorem 2.7 does not hold and, consequently,
where the theorem fails. This suggests that the faithfulness condition may be
essential for the theorem’s validity. Faithfulness ensures that the submodules
behave consistently, avoiding trivial or degenerate cases. While relaxing this
assumption would be valuable, such an example remains elusive, supporting the
conjecture that the faithfulness condition is integral to the problem’s structure.
2.9
Let Mbe a graded A-module, g∈Gsuch that every Ae-
-submodule of Mgis faithful, and Kbe a g-cl assical weakly prime A-submodule
of M. Suppose that Lis an Ae-submodule of Mg,x∈Aeand Iis an ideal of Ae.
(1) If 0=xIL ⊆K, then either xL ⊆Kor IL ⊆K.
(2) If 0=IxL ⊆K, then either xL ⊆Kor IL ⊆K.
91
J.M. HABEB—R. ABU-DAWWAS
2.10
Let Mbe a graded A-module and Kbe a g-class ical weakly
prime A-submodule of M.IfLis a faithful Ae-submodule of Mgwith LKg,
then (Kg:AeL)is a weakly prime left ideal of Ae.
P r o o f. Clearly, (Kg:AeL) is a left ideal of Ae, and since LKg,(Kg:AeL)is
a proper ideal of Ae.LetI,J be two ideals of Aesuch that 0 =IJ ⊆(Kg:AeL).
Then 0 =IJL ⊆Kg⊆K, and then by Theorem 2.7, either IL ⊆Kor JL ⊆K.
On the other hand, IL ⊆AeMg⊆Mg, and similarly, JL ⊆Mg. So, either
IL ⊆KMg=Kgor JL ⊆KMg=Kg, and thus either I⊆(Kg:AeL)or
J⊆(Kg:AeL). Hence, (Kg:AeL) is a weakly prime left ideal of Ae.
Similarly, one can prove the following:
2.11
Let Mbe a graded A-module and Kbe a g-class ical weakly
prime A-submodule of M.IfLis a faithful Ae-submodule of Mgwith LK,
then (K:AeL)is a weakly prime left ideal of Ae.
2.12
Let Mbe a graded A-module and Kbe a g-class ical weakly
prime A-submodule of M.IfAnnAe(Mg)={0},then(Kg:AeMg)is a weakly
prime left ideal of Ae.
Proof. Since Kis a g-classical weakly prime A-submodule of M,Kg=Mg,
and then (Kg:AeMg) is a proper left ideal of Ae.LetI,J be two ideals
of Aesuch that 0 =IJ ⊆(Kg:AeMg). Then IJMg⊆Kg.IfIJMg=0, then
0=IJMg⊆Kg⊆K, and then by Theorem 2.7, either IMg⊆Kor JMg⊆K.
On the other hand, IMg⊆AeMg⊆Mg, and similarly, JMg⊆Mg. So, either
IMg⊆KMg=Kgor JMg⊆KMg=Kg, and hence either I⊆(Kg:AeMg)
or J⊆(Kg:AeMg). If IJMg=0, then IJ ⊆AnnAe(Mg)={0}, a contradiction.
Thus, (Kg:AeMg) is a weakly prime left ideal of Ae.
Let M,S be two G-graded A-modules. An A-homomorphism f:M→Sis
called a graded A-homomorphism if f(Mg)⊆Sgfor all g∈G[14].
2.13
Let M, S be two G-graded A-modules and f:M→Sbe a graded
A-homomorphism.
(1) If fis injective and Kis a graded cla ssical weakly p rime A-submodule
of Swith f−1(K)=M,thenf−1(K)is a graded classi cal wea kly prime
A-submodule of M.
(2) If fis surjective and Kis a graded classical weakly prime A-submodule
of Mwith Ker(f)⊆K,thenf(K)is a graded classical wea kly p rime
A-submodule of S.
92
GRADED CLASSICAL WEAKLY PRIME SUBMODULES
Proof.
(1) By ( [15], Lemma 3.11 (1)), f−1(K) is a graded A-submodule of M.
Let x, y ∈h(A)andLbe a graded A-submodule of Msuch that
0=xAyL ⊆f−1(K). Then f(L) is a graded A-submodule of Sby ( [15 ],
Lemma 3.11 (2)) such that 0 =xAyf(L)=f(xAyL)⊆K, and then either
f(xL)=xf(L)⊆Kor f(yL)=yf(L)⊆K, which implies that either
xL ⊆f−1(K)oryL ⊆f−1(K). Hence, f−1(K) is a graded classical weakly
prime A-submodule of M.
(2) By ( [15], Lemma 3.11 (2)), f(K) is a graded A-submodule of S.Let
x, y ∈h(A)andLbe a graded A-submodule of Ssuch that 0 =xAyL ⊆
f(K). Then by ( [15], Lemma 3.11 (1)), T=f−1(L) is a graded A-
-submodule of Msuch that f(xAyT )=xAyf (T)=xAyL ⊆f(K), and
then 0 =xAyT ⊆Kas Ker(f)⊆K. So, either xT ⊆Kor yT ⊆K,and
then either xL =xf(T)=f(xT )⊆f(K)oryL =yf(T)=f(yT )⊆f(K).
Hence, f(K) is a graded classical weakly prime A-submodule of S.
Let Mbe a G-graded A-module and Tbe a graded A-submodule of M.Then
M/T is a G-graded A-module with (M/T)g=(Mg+T)/T for all g∈G.
By ([18], Lemma 2.11), if Kis an A-submodule of Mwith T⊆K,thenKis
agradedA-submodule of Mif and only if K/T is a graded A-submodule of M/T.
2.14
Let Mbe a graded A-module and T, K be proper graded A-sub-
modules of Mwith TK.IfKis a graded classical wea kly pri me A-submodule
of M,thenK/T is a graded classi cal wea kly prime A-submodule of M/T.
Proof. Let x, y ∈h(A)andL/T be a graded A-submodule of M/T such that
(0 + T)/T =(xAyL +T)/T =xAy(L+T)/T ⊆K/T.ThenLis a graded A-
-submodule of Msuch that 0 =xAyL ⊆K, and then either xL ⊆Kor yL ⊆K,
which implies that either x(L+T)/T =(xL +T)/T ⊆(K+T)/T ⊆K/T
or y(L+T)/T =(yL +T)/T ⊆(K+T)/T ⊆K/T. Hence, K/T is a graded
classical weakly prime A-submodule of M/T.
2.15
Let Mbe a graded A-module and T,K be proper graded
A-submodules of Mwith TK.IfTis a graded c las sica l weakly pri me A-
-submodule of Mand K/T is a graded c lassica l weak ly prime A-submodule
of M/T,thenKi s a graded classical weakly p rime A-submodule of M.
Proof. Let x, y ∈h(A)andLbe a graded A-submodule of Msuch that
xAyL ⊆K.IfxAyL ⊆T, then either xL ⊆TKor yL ⊆TKas
required. Suppose that xAyL T.Then
(0 + T)/T =xAy(L+T)/T =(xAyL +T)/T ⊆(K+T)/T ⊆K/T,
93
J.M. HABEB—R. ABU-DAWWAS
and then either
(xL +T)/T =x(L+T)/T ⊆K/T or (yL +T)/T =y(L+T)/T ⊆K/T,
which implies that either xL ⊆Kor yL ⊆K. Hence, Kis a graded classical
weakly prime A-submodule of M.
Graded weakly 2-absorbing ideals over non-commutative graded rings were
introduced and examined in [4]. A proper graded ideal Pof Ais said to be
raded weakly 2-absorbing if, whenever x, y, z ∈h(A) such that 0 =xAyAz ⊆P,
then xy ∈Por yz ∈Por xz ∈P. In this article, we introduce and investigate
the concept of graded weakly 2-absorbing submodules over non-commutative
graded rings as follows:
2.16
Let Mbe a graded A-module and Kbe a proper graded
A-submodule of M.Then
(1) Kis called a graded weakly 2-absorbing submodule of Mif whenever
x, y ∈h(A)andLis a graded A-submodule of Msuch that 0 =xAyL ⊆K,
then xL ⊆Kor yL ⊆Kor xAy ⊆(K:AM).
(2) Kis called a graded completely weakly 2-absorbing submodule of Mif
whenever x, y ∈h(A)andz∈h(M) such that 0 =xyz ∈K,thenxz ∈K
or yz ∈Kor xy ∈(K:AM).
Clearly, every graded classical weakly prime submodule is graded weakly
2-absorbing, and every graded completely classical weakly prime submodule is
graded completely weakly 2-absorbing. Also, if Ais a commutative graded ring
with unity, then the concepts of graded weakly 2-absorbing submodules and
graded completely weakly 2-absorbing submodules coincide. However, this will
not hold for non-commutative graded rings. For example, in Example 2.2, K
is a graded weakly 2-absorbing A-submodule of M,asitisagradedprime.
However, Kis not a graded completely weakly 2-absorbing A-submodule of M,
as 0 =xyz ∈K,xz /∈K,yz /∈Kand xy /∈(K:AM).
2.17
Let Mbe a graded A-module and Kbe a proper graded
A-submodule of M.IfKis a graded weakly 2-absorbing A-submodule of Mand
(K:AM)is a graded weakly prime ideal of A,thenKis a graded classical
weakly prime A-submodule of M.
Proof. Let x, y ∈h(A)andLbe a graded A-submodule of Msuch that 0 =
xAyL ⊆K.ThenxL ⊆Kor yL ⊆Kor xAy ⊆(K:AM)sinceKis a
graded weakly 2-absorbing A-submodule of M.IfxL ⊆Kor yL ⊆K, then the
proof is complete. Suppose that xAy ⊆(K:AM). Then as 0 =xAy,either
x∈(K:AM)ory∈(K:AM)since(K:AM) is a graded weakly prime ideal
of A, and then either xL ⊆xM ⊆Kor yL ⊆yM ⊆K. Hence, Kis a graded
classical weakly prime A-submodule of M.
94
GRADED CLASSICAL WEAKLY PRIME SUBMODULES
2.18
Let Mbe a graded A-module and Kbe a proper graded
A-submodule of M.ThenKis said to be a graded classical prime A-submodule
of Mif whenever x, y ∈h(A)andLis a graded A-submodule of Msuch that
xAyL ⊆K, then either xL ⊆Kor yL ⊆K.
Clearly, every graded classical prime submodule is a graded classical weakly
prime submodule. However, the following example shows that a graded classical
weakly prime submodule is not necessarily a graded classical prime submodule.
Example 2.19.Let A=M=M2(Z8)andG=Z4. The ring Ais G-graded as
follows
A0=Z80
0Z8,A
2=0Z8
Z80
and
A1=A3=00
00
.
Malso is a G-graded left A-module by the same graduation of A.Now,
K=00
00
is a graded classical weakly prime A-submodule of M, but Kis not a graded
classical prime A-submodule of Msince
x=20
02
∈h(A)
and L=Ax is a graded A-submodule of Mwith xAxL ⊆Kand xL K.
2.20
Let Mbe a graded A-module. If Kis a graded classical
weakly prime A-submodule of Mthat is not graded classical prime, then there
exist x, y ∈h(A)and a graded A-submodule Lof Msuch that xAyL =0,
xL Kand yL K.
Proof. Since Kis not a graded classical prime A-submodule of M,thereex-
ist x, y ∈h(A) and a graded A-submodule Lof Msuch that xAyL ⊆K,
xL Kand yL K.IfxAyL =0,thensinceKis a graded classical weakly
prime A-submodule of M,eitherxL Kor yL K, which is a contradiction.
So, xAyL =0.
2.21
Let Mbe a graded A-module, Kbe a proper graded A-
-submodule of M,Lbe a graded A-submodule of Mand x, y ∈h(A). Then
(x, y, L) is called a graded classical triple zero of Kif xAyL =0,xL Kand
yL K.
95
J.M. HABEB—R. ABU-DAWWAS
Remark 2.22
If Kis a graded classical weakly prime A-submodule of Mthat
is not graded classical prime, then by Proposition 2.20, there exists a graded clas-
sical triple zero of K. Note that, in Example 2.19, (x, x, L) is a graded classical
triple zero of K.
2.23
Let Mbe a graded A-module, Kbe a gra ded cl ass ical wea kly
prime A-submodule of Mand xAyL ⊆K,forsomex, y ∈h(A)and some graded
A-submodule Lof M.If(x, y, L)is not a graded classical triple zero of K,then
either xL ⊆Kor yL ⊆K.
Proof. If xAyL =0,thensince(x, y, L) is not a graded classical triple zero
of K,eitherxL ⊆Kor yL ⊆K.IfxAyL =0,thensinceKis a graded classical
weakly prime A-submodule of M,eitherxL ⊆Kor yL ⊆K.
2.24
Let Mbe a graded A-module and Kbe a graded classical
weakly prime A-submodule of M.If(x, y, L)is not a graded classical triple zero
of K, for all x, y ∈h(A)and all graded A-submodule Lof M,thenKis a graded
classical prime A-submodule of M.
2.25
Let Mbe a graded A-module, Kbe a gra ded cl ass ical wea kly
prime A-submodule of Mand IJL ⊆K, for some graded ideals I, J of Aand
some graded A-submodule Lof M.If(x, y, L)is not a graded classical triple zero
of K, for all x∈Ih(A)and y∈Jh(A), then for all x∈I,y∈Jand
g∈G, we have either xgL⊆Kor ygL⊆K.
Proof. Letx∈I,y∈Jand g∈G.Thenxg∈Iand yg∈Jsince Iand Jare
graded ideals, and hence xgAygL⊆IJL ⊆K.IfxgAygL=0,thensinceKis
a graded classical weakly prime A-submodule of M,eitherxgL⊆Kor ygL⊆K.
If xgAygL=0,thensince(xg,y
g,L) is not a graded classical triple zero of K,
either xgL⊆Kor ygL⊆K.
2.26
Let Mbe a graded A-module, Kbe a graded clas sical weakly
prime A-submodule of Mand IJL ⊆K, for some graded ideals I, J of Aand
some graded A-submodule Lof M.If(x, y, L)is not a graded classical triple zero
of K, for all x∈Ih(A)and y∈Jh(A), then for all g∈G, we have either
IgL⊆Kor JgL⊆K.
2.27
Let Mbe a graded A-module and Kbe a gra ded class ical weakly
prime A-submodule of M.LetLbe a graded A-submodule of Mand x, y ∈h(A).
If (x, y, L)is a graded classical triple zero of K, then the following statements
hold:
(1) xAyK =0.
(2) If y∈Ag,forsomeg∈G,thenx(K:AgM)L=0.
(3) If x∈Ag,forsomeg∈G,then(K:AgM)yL =0.
96
GRADED CLASSICAL WEAKLY PRIME SUBMODULES
(4) If x, y ∈Ag,forsomeg∈G,then(K:AgM)2L=0.
(5) If y∈Ag,forsomeg∈G,thenx(K:AgM)K=0.
(6) If x∈Ag,forsomeg∈G,then(K:AgM)yK =0.
(7) If x, y ∈Ag,forsomeg∈G,then(K:AgM)2K=0.
Proof.
(1) Suppose that xAyK = 0. Then there exists z∈Ksuch that xAyz =0,
andthenthereexistsg∈Gsuch that xAyzg= 0, and hence xAyAzg=0.
Note that zg∈Kas Kis a graded A-submodule. Let T=L+Azg.
Then Tis a graded A-submodule of Msuch that 0 =xAyT ⊆K,and
then either xT ⊆Kor yT ⊆K, which implies that either xL ⊆xT ⊆K
or yL ⊆yT ⊆K, which is a contradiction. Thus xAyK =0.
(2) Suppose that x(K:AgM)L= 0. Then there exists s∈(K:AgM)such
that xsL =0,andthenxAsL = 0, and hence 0 =xA(y+s)L⊆K.
So, either xL ⊆Kor (y+s)L⊆K, and then either xL ⊆Kor yL ⊆K,
which is a contradiction. Hence, x(K:AgM)L=0.
(3) Similar to (2).
(4) Suppose that (K:AgM)2L= 0. Then there exist r, s ∈(K:AgM)
such that rsL =0,andthenrAsL = 0, and hence by (2) and (3),
0=(x+r)A(y+s)L⊆K. So, either (x+r)L⊆Kor (y+s)L⊆K,
andtheneitherxL ⊆Kor yL ⊆K, which is a contradiction. Hence,
(K:AgM)2L=0.
(5) Suppose that x(K:AgM)K= 0. Then there exists s∈(K:AgM)such
that xsK =0,andthenxAsK = 0, and hence by (1), xA(y+s)K=0,
which implies that xA(y+s)z= 0, for some z∈K, and so there exists
h∈Gsuch that xA(y+s)zh=0.Notethatzh∈Kas Kis a graded A-
-submodule. Let T=L+Azh.ThenTis a graded A-submodule of Msuch
that 0 =xA(y+s)T⊆K, and then either xT ⊆Kor (y+s)T⊆K,and
hence either xT ⊆Kor yT ⊆K, which implies that either xL ⊆xT ⊆K
or yL ⊆yT ⊆K, which is a contradiction. Hence, x(K:AgM)K=0.
(6) Similar to (5).
(7) Suppose that (K:AgM)2K= 0. Then there exist r, s ∈(K:AgM)and
z∈Ksuch that rsz = 0, and then there exists h∈Gsuch that rszh=0.
Note that zh∈Kas Kis a graded A-submodule. Let T=L+Azh.Then
Tis a graded A-submodule of Msuch that 0 =(x+r)A(y+s)T⊆K
by (2) and (3), and then either (x+r)T⊆Kor (y+s)T⊆K, and hence
either xT ⊆Kor yT ⊆K, which implies that either xL ⊆xT ⊆Kor
yL ⊆yT ⊆K, which is a contradiction. Hence, (K:AgM)2K=0.
97
J.M. HABEB—R. ABU-DAWWAS
2.28
Let Mbe a graded A-module and Kbe a graded classical
weakly prime A-submodule of M. If there exists a graded classical triple zero
(x, y, L)of Kwith x, y ∈Ae,then(K:AeM)3⊆AnnAe(M).
P r o o f. By Theorem 2.27 (7), (K:AeM)2K=0.So,
(K:AeM)3=(K:AeM)2(K:AeM)
⊆(K:AeM)2K:AeM
=(0:
AeM) = Ann
Ae
(M).
As a consequence of Proposition 2.28, we have the following:
2.29
Let Mbe a faithful graded A-module and Kbe a graded
clas sical weakly prime A-submodule of M. If there exists a graded classical triple
zero (x, y, L)of Kwith x, y ∈Ae,then(K:AeM)3=0.
Also, as a consequence of Theorem 2.27 (7), we have the following:
2.30
Let Mbe a graded A-module and Kbe a faithful graded
clas sical weakly prime A-submodule of M. If there exists a graded classical triple
zero (x, y, L)of Kwith x, y ∈Ag,forsomeg∈G,then(K:AgM)2=0.
Furthermore, as a consequence of Theorem 2.27 (4), we have the following:
2.31
Let Mbe a graded A-module and Kbe a graded classi-
cal weakly pri me A-submodule of M. If there exists a graded classical triple
zero (x, y, L)of Kwith x, y ∈Ag,forsomeg∈G,andLis faithful, then
(K:AgM)2=0.
Let Mand Sbe two G-graded A-modules. Then M×Sis a G-graded
A-module by (M×S)g=Mg×Sg, for all g∈G[14]. Moreover, N=K×Tis
agradedA-submodule of M×Sif and only if Kis a graded A-submodule of M
and Tis a graded A-submodule of S([18, Lemma 2.10 and Lemma 2.12]).
2.32
Let Mand Sbe two G-graded A-modules. If K×Sis a graded
clas sical weakly prime A-submodule of M×S,thenKis a graded classica l weakly
prime A-submodule of M.
Proof. Let x, y ∈h(A)andLbe a graded A-submodule of Msuch that
0=xAyL ⊆K.ThenL×{0}is a graded A-submodule of M×Ssuch that
(0,0) =xAy(L×{0})⊆K×S, and then either x(L×{0})⊆K×Sor
y(L×{0})⊆K×S, and hence either xL ⊆Kor yL ⊆K.ThusKis a graded
classical weakly prime A-submodule of M.
2.33
Let Mand Sbe two G-graded A-modules and K×Sis a graded
clas sical weakly pri me A-submodule of M×S.If(x, y, L)is a graded classical
triple zero of K,thenxAy ⊆AnnA(S).
98
GRADED CLASSICAL WEAKLY PRIME SUBMODULES
P r o o f. Suppose that xAy AnnA(S). Then there exists s∈Ssuch that
xAys = 0, and then there exists g∈Gsuch that xAysg=0.Now,L×Asg
is a graded A-submodule of M×Ssuch that (0,0) =xAy(L×Asg)⊆K×S,
so either x(L×Asg)⊆K×Sor y(L×Asg)⊆K×S, and hence either xL ⊆K
or yL ⊆K, which is a contradiction. Thus xAy ⊆AnnA(S).
We close this section by introducing a nice result concerning graded weakly
prime submodules over graded multiplication modules (Theorem 2.35). A graded
A-module Mis called a graded multiplication if for every graded A-submodule
Kof M,K=IM, for some graded ideal Iof A. In this case, it is known that
I=(K:AM). Graded multiplication modules were first introduced and studied
by Escoriza and Torrecillas in [7], and further results were obtained by several
authors, see for example [6,12].
2.34
Let Mbe a graded A-module. Then every graded maximal
A-submodule of Mis graded prime.
Proof. LetKbe a graded maximal A-submodule of M. Suppose that IL ⊆K,
for some graded ideal Iof Aand some graded A-submodule Lof M.
Assume that LK.SinceKis a graded maximal A-submodule of M,L+K=M,
and hence IM =IL +IK ⊆K, which implies that I⊆(K:AM). Hence, K
is a graded prime A-submodule of M.
2.35
Let Mbe a graded multiplication A-module. If every proper
graded A-submodule of Mis graded weakly prime, then Mhas at most two
graded maximal A-submodules.
Proof. LetX,Yand Zbe three distinct graded maximal A-submodules of M.
Since Mis a graded multiplication, X=IM, for some graded ideal Iof A.
If IY =0,thenIY ⊆Z, and since Zis graded weakly prime by Lemma 2.34,
either Y⊆Zor X=IM ⊆Z, which is a contradiction. So, IY =0.
Now, IY ⊆Yand IY ⊆IM =X,so0=IY ⊆XY, and since XYis
graded weakly prime by assumption, either Y⊆XYor X=IM ⊆XY,
which implies that either Y⊆Xor X⊆Y, which is a contradiction. Hence, M
has at most two graded maximal A-submodules.
2.36
Let Mbe a graded multiplication A-module such that every
proper graded A-submodule of Mis graded weakly prime. If X=IM and Y=JM
are two distinct graded A-submodules of M, for some graded ideals I,J of A,then
either Xand Yare comparable by inclusion or IY =JX =0.Inparticular,ifX
and Yare two distinct graded maximal A-submodules of M,thenIY =JX =0.
99
J.M. HABEB—R. ABU-DAWWAS
3. Graded classical weakly prime submodules
over Duo graded rings
In this section, we study graded classical weakly prime submodules over
Duo graded rings. A ring Ais said to be a left Duo ring if every left ideal
of Ais a two sided ideal [9, 13]. It is obvious that if Ais a left Duo ring, then
xA ⊆Ax, for all x∈A.
3.1
Let Abe a left Duo graded ring, Mbe a graded A-module and
Kbe a graded classical weakly prime A-submodule of M.Ifx, y ∈h(A)and
m∈h(M)such that 0=xym ∈K, then either xm ∈Kor ym ∈K.
Proof. Since Ais a left Duo ring, Axy =AxAyA,andthenAm is a graded
A-submodule of Msuch that 0 =xAyAm ⊆K, and so either xAm ⊆Kor
yAm ⊆K, that implies either xm ∈Kor ym ∈K.
Theorem 3.1 states that every graded classical weakly prime submodule
of a graded module over a left Duo graded ring is graded completely classical
weakly prime. As a consequence of Theorem 3.1, we have the following:
3.2
Let Abe a left Duo graded ring, Mbe a graded A-module
and Kbe a graded cla ssi cal weakly p rime A-submodule of M.Ifx, y ∈h(A)and
m∈h(M)such that xym ∈Kand (x, y, Am)is not graded classical triple zero
of K, then either xm ∈Kor ym ∈K.
Let Mbe an A-module and Kbe an A-submodule of M.ThenKis said
to be an u-submodule if whenever K⊆n
j=1 Kj, for some A-submodules Kj’s
of M,thenK⊆Kj, for some 1 ≤j≤n.Mis said to be an u-module if every
A-submodule of Mis an u-submodule [16]. Let Mbe an A-module, Kbe an
A-submodule of Mand r∈A.Then(K:Mr)={m∈M:rm ∈K}is
an A-submodule of Mcontaining K.
3.3
Let Mbe a graded A-module such that Aeis a left Duo ring
and Mgis an u-module over Ae,forsomeg∈G. Suppose that Kis a graded
A-submodule of Msuch that Kg=Mg. Consider the following statements:
(1) Kis a g-cla ssical weakly prime A-submodule of M.
(2) If x, y ∈Aeand m∈Mgsuch that 0=xym ∈K, then either xm ∈Kor
ym ∈K.
(3) Fo r a l l x, y ∈Ae,(K:Mgxy)=(0:
Mgxy)or (K:Mgxy)=(K:Mgx)or
(K:Mgxy)=(K:Mgy).
100
GRADED CLASSICAL WEAKLY PRIME SUBMODULES
(4) If x, y ∈Aeand Lis an Ae-submodule of Mgsuch that 0=xyL ⊆K,
then either xL ⊆Kor yL ⊆K.
(5) If x∈Aeand Lis an Ae-submodule of Mgsuch that xL K, then either
(K:AexL)=(0:
AexL)or (K:AexL)=(K:AeL).
(6) If x∈Ae,Iis an ideal of Aeand Lis an Ae-submodule of Mgsuch that
0=IxL ⊆K, then either IL ⊆Kor xL ⊆K.
(7) If Iis an ideal of Aeand Lis an Ae-submodule of Mgsuch that IL K,
then either (K:AeIL)=(0:
AeIL)or (K:AeIL)=(K:AeL).
Then (1) ⇒(2) ⇒(3) ⇒(4) ⇒(5) ⇒(6) ⇒(7).
Proof.
(1) ⇒(2):Similar to proof of Theorem 3.1.
(2) ⇒(3):Let x, y ∈Aeand m∈(K:Mgxy). Then xym ∈K.Ifxym =0,
then m∈(0 :Mgxy). If xym =0,thenby(2),eitherxm ∈Kor ym ∈K,
and so either m∈(K:Mgx)orm∈(K:Mgy). Hence,
(K:Mgxy)⊆(0 :Mgxy)(K:Mgx)(K:Mgy),
and then (K:Mgxy)⊆(0 :Mgxy)or(K:Mgxy)⊆(K:Mgx)or
(K:Mgxy)⊆(K:Mgy)asMgis an u-module over Ae, and hence
the result holds as Aeis a left Duo ring.
(3) ⇒(4):Clearly, L⊆(K:Mgxy)andK(0 :Mgxy), so by (3), either
L⊆(K:Mgx)orL⊆(K:Mgy), and then either xL ⊆Kor yL ⊆K.
(4) ⇒(5):Let r∈(K:AexL). Then rxL⊆K.IfrxL =0, then r∈(0 :AexL).
If rxL=0, then by (4), rL⊆K.So,
(K:AexL)⊆(0 :AexL)(K:AeL).
Hence, the result holds as Aeis a left Duo ring.
(5) ⇒(6):Clearly, I⊆(K:AexL)andI(0 :AexL), so by (5), either
xL ⊆Kor I⊆(K:AeL), and then either xL ⊆Kor IL ⊆K.
(6) ⇒(7):Let x∈(K:AeIL). Then xIL ⊆K.IfxI L =0,then
x∈(0 :AeIL). If xIL =0,thenasAeis a left Duo ring, 0 =IxL ⊆K,
and then by (6), xL ⊆K, and hence x∈(K:AeL). Thus
(K:AeIL)⊆(0 :AeIL)(K:AeL).
So, the result holds as Aeis a left Duo ring.
101
J.M. HABEB—R. ABU-DAWWAS
Remark 3.4
The assumptions of Theorem 3.3 may seem restrictive at first
glance, particularly because finding an explicit example of a graded ring that
satisfies all the prerequisites can be challenging. The conditions are specifically
designed to capture certain algebraic properties that, while not common in ele-
mentary examples, arise in more specialized contexts, such as in some construc-
tions involving homogeneous elements or in graded structures arising from al-
gebraic geometry or commutative algebra. Despite extensive efforts, a concrete
example that satisfies all of the theorem’s assumptions has not been found.
This suggests that the class of graded rings that fulfill these conditions may
be more specialized or less straightforward to construct explicitly. However,
certain well-known classes of graded rings, such as graded polynomial rings
or quotient rings of graded algebras, may offer fruitful directions to explore.
Further research could involve either relaxing some of the theorem’s assump-
tions or developing a more detailed classification of graded rings that fit the
prerequisites. For now, the theoretical framework provided by Theorem 3.3
remains important for understanding the broader structural properties of graded
rings under these conditions.
Let Abe a left Duo graded ring. If Ais a graded ring, then clearly,
Grad
A(I)={x∈A:∀g∈G, ∃ng∈Ns.t xng
g∈I}
is a graded ideal of Acontaining I.Evidently,ifx∈h(A), then x∈GradA(I)
if and only if xn∈Ifor some n∈N.
3.5
Let Mbe a graded A-module such that Aeis a left Duo
ring and Kbe a graded classical wea kly prime A-submodule of M.Ifthereexists
a graded classical triple zero (x, y, L)of Kwith x, y ∈Ae,then
Grad
AeAnn
Ae
(M)=Grad
Ae(K:AeM).
P r o o f. By Proposition 2.28,
K:AeM⊆Grad
AeAnn
Ae
(M),
and then
Grad
Ae
((K:AeM)) ⊆Grad
AeGrad
AeAnn
Ae
(M)
=Grad
AeAnn
Ae
(M).
On the other hand, since
Ann
Ae
(M)⊆(K:AeM),Grad
AeAnn
Ae
(M)⊆Grad
Ae
((K:AeM)).
Hence,
Grad
AeAnn
Ae
(M)=Grad
Ae
((K:AeM)).
102
GRADED CLASSICAL WEAKLY PRIME SUBMODULES
Acknowledgement
The authors would like to express sincere gratitude to the
anonymous referees for their insightful comments and suggestions that greatly
contributed to improving the quality of this article. Their thorough review and
valuable feedback were instrumental in shaping the final version of this work.
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https://mathscinet.ams.org/mathscinet/relay-station?mr=4199177
Received June 18, 2023
Revised September 26, 2024
Accepted October 3, 2024
Publ. online November 30, 2024
Jebrel M. Habeb
Rashid Abu-Dawwas
Department of Mathematics
Faculty of Science
Yarmouk University
21163-Irbid
JORDAN
E-mail: jhabeb@yu.edu.jo
rrashid@yu.edu.jo
104
