Graded gw-Prime Submodules

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Graded gw-Prime Submodules
RASHID ABU-DAWWAS
Department of Mathematics
Yarmouk University
Irbid, Jordan
rrashid@yu.edu.jo
Abstract—Let Gbe a group with identity e,Ra commutative
G-graded ring with unity 1and MaG-graded R-module. In this
article, graded gw-prime submodules of Mare defined. This class
of graded submodules is a generalization of graded weakly prime
submodules. A graded R-module Mis said to be graded gw-
prime if whenever Lis a graded R-submodule of Mand x, y ∈
h(R)such that xyL ={0}, then either x2L={0}or y2L={0}.
A graded R-submodule Nof Mis said to be graded gw-prime
if M/N is a graded gw-prime R-module, i.e., if whenever Lis a
graded R-submodule of Mand x, y ∈h(R)such that xyL ⊆N,
then either x2L⊆Nor y2L⊆N. Also, we introduce the concept
of graded valuation modules; let RbeaG-graded domain with
quotient field Fand MaG-graded torsion free R-module. For
b=x
y∈Fwhere x, y ∈h(R)and for s∈h(M), we write bs ∈M
if there exists t∈h(M)such that xs =yt. Then Mis said to be a
graded valuation R-module if for each b=x
y∈K(x, y ∈h(R)),
we have either bM ⊆Mor b−1M⊆M. After studying general
properties of graded gw-prime submodules, their relation with
graded valuation modules are examined.
Index Terms—Graded gw-prime modules, graded gw-prime
submodules, graded weakly prime submodules, graded valuation
modules.
I. INTRODUCTION
Let Gbe a group with identity e,Ra commutative G-
graded ring with unity 1and MaG-graded R-module. Graded
prime submodules have been introduced by S. E. Atani in
[9] and then studied extensively by many authors as ( [3],
[4], [8], [16]). A graded R-submodule Nof Mis said to be
graded prime if whenever r∈h(R)and x∈h(M)such
that rx ∈N, then either x∈Nor rM ⊆N. In recent
years, several generalizations of graded prime submodules are
obtained, see ( [1], [5], [6], [7], [11], [12]). Among those gen-
eralizations, the concept of graded weakly prime submodules
distinguishes itself with its useful properties. Graded weakly
prime submodules are first introduced by S. E. Atani in [10].
A graded R-submodule Nof Mis said to be graded weakly
prime if whenever Lis a subset of Mand a, b ∈h(R)such
that abL ⊆N, then either aL ⊆Nor bL ⊆N. For a
generalization of graded weakly prime submodules, the reader
is referred to [2].
In this article, we follow [13] to define graded gw-prime
submodules. A graded R-module Mis said to be graded gw-
prime if whenever Lis a graded R-submodule of Mand x, y∈
h(R)such that xyL ={0}, then either x2L={0}or y2L=
{0}. A graded R-submodule Nof a graded R-module Mis
said to be graded gw-prime if M/N is a graded gw-prime
R-module. This definition can be explained as such: A graded
R-submodule Nof a graded R-module Mis said to be graded
gw-prime if whenever Lis a graded R-submodule of Mand
x, y ∈h(R)such that xyL ⊆N, then either x2L⊆Nor
y2L⊆N.
Also, we introduce the concept of graded valuation modules;
let Rbe a G-graded domain with quotient field Fand Ma
G-graded torsion free R-module. For b=x
y∈Fwhere x, y ∈
h(R)and for s∈h(M), we write bs ∈Mif there exists
t∈h(M)such that xs =yt. Then Mis said to be a graded
valuation R-module if for each b=x
y∈K(x, y ∈h(R)), we
have either bM ⊆Mor b−1M⊆M.
General properties of graded gw-prime submodules are
investigated. Also, the relation between graded gw-prime
submodules and graded valuation modules is examined.
We start by recalling some background material. Let Gbe
a group with identity eand Ra commutative ring with a
nonzero unity 1. Then Ris said to be G-graded if R=
g∈G
Rg
with RgRh⊆Rgh for all g, h ∈Gwhere Rgis an additive
subgroup of Rfor all g∈G. The elements of Rgare
called homogeneous of degree g. Consider supp(R, G)=
{g∈G:Rg=0}.Ifx∈R, then xcan be written as 
g∈G
xg,
where xgis the component of xin Rg. Moreover, Reis a
subring of Rand 1∈Reand h(R)= 
g∈G
Rg.
Let Ibe an ideal of a graded ring R. Then Iis said to be
graded ideal if I=
g∈G
(I∩Rg), i.e., for x∈I,x=
g∈G
xg
where xg∈Ifor all g∈G. An ideal of a graded ring need
not be graded (see [15]).
Assume that Mis an R-module. Then Mis said to be G-
graded if M=
g∈G
Mgwith RgMh⊆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. Also,
we consider supp(M,G)={g∈G:Mg=0}. It is clear
that Mgis an Re-submodule of Mfor all g∈G. Moreover,
h(M)= 
g∈G
Mg.
Let Nbe an R-submodule of a graded R-module M. Then
Nis said to be graded R-submodule if N=
g∈G
(N∩Mg),
i.e., for x∈N,x=
g∈G
xgwhere xg∈Nfor all g∈G.An
R-submodule of a graded R-module need not be graded (see
1
2019 International Conference on Mathematics and Computers in Science and Engineering (MACISE)
978-1-5386-9204-2/19/$31.00 ©2019 IEEE
DOI 10.1109/MACISE.2019.00007

[15]).
If Mis a G-graded R-module and Nan R-submodule
of M, then M/N is a G-graded R-module by (M/N)g=
(Mg+N)/N for all g∈G.
Lemma 1.1: ( [14], Lemma 2.1) Let Mbe a graded R-
module.
1) If Iand Jare graded ideals of R, then I+Jand IJ
are graded ideals of R.
2) If Nand Kare graded R-submodules of M, then N+K
and NKare graded R-submodules of M.
3) If Nis a graded R-submodule of M,r∈h(R),x∈
h(M)and Iis a graded ideal of R, then Rx,IN and
rN are graded R-submodules of M.
Lemma 1.2: ( [16], Lemma 1) Let Mbe a graded R-module
and Na graded R-submodule of M. Then (N:RM)=
{r∈R:rM ⊆N}is a graded ideal of R.
II. GRADED gw-PRIME SUBMODULES
In this section, we introduce and study graded gw-prime
submodules.
Definition 2.1: Let Mbe a graded R-module. Then Mis
said to be graded gw-prime if whenever Lis a graded R-
submodule of Mand x, y ∈h(R)such that xyL ={0},
then either x2L={0}or y2L={0}. Let Nbe a graded
R-submodule of M. Then Nis said to be graded gw-prime
if M/N is a graded gw-prime R-module, i.e., if whenever L
is a graded R-submodule of Mand x, y ∈h(R)such that
xyL ⊆N, then either x2L⊆Nor y2L⊆N.
Theorem 2.2: Let Mbe a graded R-module. Then Mis a
graded gw-prime R-module if and only if whenever x, y ∈
h(R)and s∈h(M)such that xys =0, then either x2s=0
or y2s=0.
Proof 1: Suppose that Mis a graded gw-prime R-module.
Let x, y ∈h(R)and s∈h(M)such that xys =0. By Lemma
1.1, L=Rs is a graded R-submodule of Msuch that xyL =
{0}and then either x2L={0}or y2L={0}which implies
that either x2s=0or y2s=0. Conversely, Let Lbe a graded
R-submodule of Mand x, y ∈h(R)such that xyL ={0}.
Then for each s∈L,wehavexys =0. Since s∈Land Lis
a graded R-submodule of M,s=
g∈G
sgwhere sg∈Lfor all
g∈G.Now,
g∈G
xysg=xy 
g∈G
sg=xys =0∈{0}. Since
xysg∈h(M)for all g∈Gand {0}is a graded R-submodule
of M,xysg=0for all g∈G. By assumption, for all g∈G,
either x2sg=0or y2sg=0. Let N=
sg∈L:x2sg=0
{sg}
and K=
sg∈L:y2sg=0
{sg}. Then Nand Kare graded R-
submodules of Mwith L=NK. So, we have either N⊆
Kor K⊆N. Suppose that N⊆K. Then L=Kand then
y2sg=0for all g∈Gwhich implies that y2s=y2
g∈G
sg=

g∈G
y2sg=0for all s∈Land hence y2L={0}. Similarly,
if K⊆N, then x2L={0}. Thus, Mis a graded gw-prime
R-module.
Corollary 2.3: Let Mbe a graded R-module and Ka
graded R-submodule of M. Then Kis a graded gw-prime
R-submodule of Mif and only if whenever x, y ∈h(R)
and s∈h(M)such that xys ∈K, then either x2s∈K
or y2s∈K.
Let Rbe a G-graded ring and Ia graded ideal of R. Then
the graded radical of Iis denoted by Gr(I)and it is defined to
be the set of all x∈Rsuch that for each g∈G, there exists a
positive integer ngsatisfies xng
g∈I. One can see that if xis a
homogeneous element, then x∈Gr(I)if and only if xn∈I
for some positive integer n. Graded prime ideals have been
introduced and studied by M. Refai, M. Hailat and S. Obiedat
in [17]. A proper graded ideal Pof a graded ring Ris said
to graded prime if whenever a, b ∈h(R)such that ab ∈P,
then either a∈Por b∈P. Let Nbe a graded R-submodule
of a graded R-module M. Then by Lemma 1.2, (N:RM)is
a graded ideal of R. The next theorem proves that if Nis a
graded gw-prime R-submodule of M, then Gr((N:RM)) is
a graded prime ideal of R.
Theorem 2.4: Let Nbe a graded R-submodule of a graded
R-module M.IfNis a graded gw-prime R-submodule of M,
then Gr((N:RM)) is a graded prime ideal of R.
Proof 2: Let a, b ∈h(R)such that ab ∈Gr((N:RM)).
Then anbnM⊆Nfor some positive integer n. Since Nis
graded gw-prime, either a2nM⊆Nor b2nM⊆Nwhich
implies that either a∈Gr((N:RM)) or b∈Gr((N:RM)).
Hence, Gr((N:RM)) is a graded prime ideal of R.
For G-graded R-modules M(1) and M(2),anR-
homomorphism f:M(1) →M(2) is said to be graded R-
homomorphism if f(M(1)
g)⊆M(2)
gfor all g∈G( [15]).
Theorem 2.5: Let f:M(1) →M(2) be a graded R-
homomorphism.
1) If Nis a graded gw-prime R-submodule of M(2), then
f−1(N)is a graded gw-prime R-submodule of M(1).
2) If fis surjective and Na graded gw-prime R-
submodule of M(1) such that Ker(f)⊆N, then f(N)
is a graded gw-prime R-submodule of M(2).
Proof 3:
1) Let a, b ∈h(R)and m∈h(M(1) )such that abm ∈
f−1(N). Then f(m)∈h(M(2))such that abf (m)=
f(abm)∈N. Since Nis graded gw-prime, either
a2f(m)∈Nor b2f(m)∈Nwhich implies that either
a2m∈f−1(N)or b2m∈f−1(N). Hence, f−1(N)is
a graded gw-prime R-submodule of M(1).
2) Let a, b ∈h(R)and m∈h(M(2) )such that abm ∈
f(N). Then there exists t∈Nsuch that f(t)=abm.
Since fis surjective, there exists s∈h(M(1))such that
f(s)=mand then f(abs)=f(t)which implies that
t−abs ∈Ker(f)⊆Nand hence abs ∈N. Since
Nis graded gw-prime, either a2s∈Nor b2s∈N
and then either a2m=a2f(s)=f(a2s)∈f(N)or
b2m∈f(N). Hence, f(N)is a graded gw-prime R-
submodule of M(2).
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Corollary 2.6: Let Mbe a graded R-module and N,K
graded R-submodules of Msuch that N⊆K. Then Kis a
graded gw-prime R-submodule of Mif and only if K/N is
a graded gw-prime R-submodule of M/N.
Proof 4: Apply Theorem 2.5 on the graded R-
homomorphism f:M→M/N defined by f(x)=x+N.
For G-graded R-modules M(1) and M(2),M(1) ×M(2) is
a graded R-module by (M(1) ×M(2))g=M(1)
g×M(2)
gfor
all g∈G( [15]).
Corollary 2.7: Let M(1) and M(2) be G-graded R-modules,
Na proper graded R-submodule of M(1) and Ka proper
graded R-submodule of M(2).
1) N×M(2) is a graded gw-prime R-submodule of
M(1) ×M(2) if and only if Nis a graded gw-prime
R-submodule of M(1).
2) If N×Kis a graded gw-prime R-submodule of M(1) ×
M(2), then Nis a graded gw-prime R-submodule of
M(1) and Ka graded gw-prime R-submodule of M(2).
Proof 5: Apply Theorem 2.5 on the projection f1:M(1) ×
M(2) →M(1) and f2:M(1) ×M(2) →M(2).
Theorem 2.8: Let M(1) be a G-graded R(1)-module, M(2) a
G-graded R(2)-module, Na graded R(1) -submodule of M(1),
Ka graded R(2)-submodule of M(2) ,R=R(1) ×R(2) and
M=M(1) ×M(2).
1) N×M(2) is a graded gw-prime R-submodule of Mif
and only if Nis a graded gw-prime R(1)-submodule of
M(1).
2) If N×Kis a graded gw-prime R-submodule of M,
then Nis a graded gw-prime R(1)-submodule of M(1)
and Ka graded gw-prime R(2)-submodule of M(2) .
Proof 6:
1) Suppose that N×M(2) is a graded gw-prime R-
submodule of M. Let x, y ∈h(R(1))and t∈h(M(1))
such that xyt ∈N. Then (x, 1),(y, 1) ∈h(R)and
(t, 0) ∈h(M)such that (x, 1)(y, 1)(t, 0) ∈N×M(2)
and then either (x2t, 0) ∈N×M(2) or (y2t, 0) ∈
N×M(2) which implies that either x2t∈Nor y2t∈N.
Hence, Nis a graded gw-prime R(1)-submodule of
M(1). Conversely, let (x1,x
2),(y1,y
2)∈h(R)and
(s, t)∈h(M)such that (x1,x
2)(y1,y
2)(s, t)∈N×
M(2). Then x1,y
1∈h(R(1))and s∈h(M(1) )such
that x1y1s∈Nand then either x2
1s∈Nor y2
1s∈N
which implies that (x1,x
2)2(s, t)∈N×M(2) or
(y1,y
2)2(s, t)∈N×M(2). Hence, N×M(2) is a graded
gw-prime R-submodule of M.
2) Similar to (1).
A graded R-submodule Nof a graded R-module Mis said
to be homogeneous cyclic if N=Rx for some x∈h(M).
Theorem 2.9: Let Mbe a graded R-module. Then every
homogeneous cyclic graded R-submodule of Mis gw-prime
if and only if every graded R-submodule of Mis gw-prime.
Proof 7: Suppose that every homogeneous cyclic graded
R-submodule of Mis gw-prime. Let Nbe a graded R-
submodule of M. Assume that a, b ∈h(R)and t∈h(M)
such that abt ∈N. Then abt =nfor some n∈N. Since
a, b ∈h(R)and t∈h(M),there exist g, h, z ∈Gsuch
that a∈Rg,b∈Rhand t∈Mzand then abt ∈Mghz
which implies that abt =(abt)ghz =nghz and hence
abt ∈Rnghz. Since Rnghz is a gw-prime R-submodule of
M, either a2t∈Rnghz or b2t∈Rnghz. Since n∈Nand
Nis graded, nghz ∈Nand then Rnghz ⊆N. So, we have
either a2t∈Nor b2t∈N. Hence, Nis a graded gw-prime
R-submodule of M. The converse is clear.
Definition 2.10: Let Rbe a G-graded domain with quotient
field Fand MaG-graded torsion free R-module. For b=x
y∈
Fwhere x, y ∈h(R)and for s∈h(M), we write bs ∈M
if there exists t∈h(M)such that xs =yt. Then Mis said
to be a graded valuation R-module if for each b=x
y∈F
(x, y ∈h(R)), we have either bM ⊆Mor b−1M⊆M.
Theorem 2.11: Let Rbe a G-graded domain and MaG-
graded torsion free R-module. If every homogeneous cyclic
graded R-submodule of Mis graded gw-prime, then Mis a
graded valuation R-module.
Proof 8: Let Fbe the quotient field of Rand b=x
y∈F
where x, y ∈h(R). Suppose that m∈M. Then m=
g∈G
mg
where mg∈Mgfor all g∈G.Ifbmg∈Mfor all g∈G, then
bm ∈Mand hence bM ⊆M. Assume that bmg/∈Mfor
some g∈G. Then for all t∈h(M),wehavexmg=yt.
Now, N=Rxymgis a homogeneous cyclic graded R-
submodule and hence graded gw-prime R-submodule. So,
either Rx2mg⊆Nor Ry2mg⊆N.IfRx2mg⊆N,
then there exists c∈h(R)such that x2mg=cxymg. Since
Mis torsion free and Ris a domain, xmg=cymgwhich
is a contradiction. Hence, Ry2mg⊆Nwhich implies that
y=sx for some s∈h(R)and then b−1=y
x=sand hence
b−1M=sM ⊆M. Therefore, Mis a graded valuation R-
module.
We close our article with an example shows that the
converse of Theorem 2.11 is not true in general.
Example 2.12: Let R=Z,M=Z(p)×Z(p)where Z(p)=
x
y∈Q:pdoes not divide ywhere pis a prime number. If
G=Z2(the group of integers modulo 2), then Ris trivially
G-graded by R0=Zand R1={0}. Also, Mis G-graded by
M0=Z(p)×{0}and M1={0}×Z(p). Let b∈Q. Then b=
α
βfor some α, β ∈R0⊆h(R)with gcd(α, β)=1. Assume
that m∈M. Then m=(
x1
y1,x2
y2)for some x1,x
2,y
1,y
2∈Q
such that pdoes not divide y1and does not divide y2. Note
that m=m1+m2where m1=(
x1
y1,0) ∈M1⊆h(M)and
m2=(0,x2
y2)∈M2⊆h(M).Ifbm1∈Mand bm2∈M,
then bm ∈Mand then bM ⊆M. Suppose that bm1/∈M.
Then pdivides βand since gcd(α, β )=1,pdoes not divide
αand hence b−1m1∈Mand b−1m2∈Mwhich implies that
b−1m∈M. Similarly, if bm2/∈M. Hence, b−1M⊆M. So,
Mis a graded valuation R-module. Now, N=6
1×0=
(6
1,0)is a homogeneous cyclic graded R-submodule of M
such that 2,3∈h(R)and (1
1,0) ∈h(M)with (2)(3)(1
1,0) ∈
Nbut 4( 1
1,0) /∈Nand 9( 1
1,0) /∈N. Hence, Nis not a graded
gw-prime R-submodule of M.
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