()


@ Appl. Gen. Topol. 14, no. 2 (2013), 159-169doi:10.4995/agt.2013.1586
c© AGT, UPV, 2013

Zariski topology on the spectrum of graded

classical prime submodules

Ahmad Yousefian Darani
a
and Shahram Motmaen

b

a
Department of Mathematics and Applications, Faculty of Mathematical Sciences, University of

Mohaghegh Ardabili, 56199-11367, Ardabil, Iran. (yousefian@uma.ac.ir, youseffian@gmail.com)
b
Young Researchers Club, Ardabil Branch Islamic Azad University, Ardabil, Iran. (sh.motmaen@gmail.com)

Abstract

Let R be a G-graded commutative ring with identity and let M be a

graded R-module. A proper graded submodule N of M is called graded

classical prime if for every a, b ∈ h(R), m ∈ h(M), whenever abm ∈ N,
then either am ∈ N or bm ∈ N. The spectrum of graded classical

prime submodules of M is denoted by Cl.Specg(M). We topologize
Cl.Specg(M) with the quasi-Zariski topology, which is analogous to
that for Specg(R).

2010 MSC: 13A02, 16W50.

Keywords: Graded prime ideal, Zariski topology, quasi-Zariski topology.

1. Introduction

Recently many authors have been interested in equip algebraic structures
with Zariski topology (cf. [4, 11, 12]). A grading on a ring and its mod-
ules usually aids computations by allowing one to focus on the homogeneous
elements, which are presumably simpler or more controllable than random el-
ements. However, for this to work one needs to know that the constructions
being studied are graded. One approach to this issue is to redefine the con-
structions entirely in terms of the category of graded modules and thus avoid
any consideration of non-graded modules or non-homogeneous elements; Sharp
gives such a treatment of attached primes in [15]. Unfortunately, while such an
approach helps to understand the graded modules themselves, it will only help

Received August 2012 – Accepted March 2013

http://dx.doi.org/10.4995/agt.2013.1586


A. Yousefian Darani and S. Motmaen

to understand the original construction if the graded version of the concept
happens to coincide with the original one. Therefore, notably, the study of
graded modules is very important.

Our main purpose is to study some new classes of graded submodules of
graded modules and endow these classes of submodules with quasi-Zariski
topology. Zariski topology on the prime spectrum of a module over a com-
mutative ring have been already studied in [11, 12]. Moreover some topologies
on the spectrum of graded prime submodules of a graded module have been
studied in [16]. Therefore these results will be used in order to obtain the main
aims of this paper.

The organization of this paper is as follows: In section 2 we recollect the
results concerning the topologies on the prime spectrum of a module over a
commutative ring. Moreover we remind the notation and the elemental prop-
erties about graded modules and rings that we will use in this paper. In section
3 we introduce and study the concept of graded classical prime submodules and
define the quasi-Zariski topology on the spectrum of all graded classical prime
submodules of a graded module.

2. preliminaries

In this section, we recall some definitions and notations used throughout.
Throughout this paper all rings are commutative with a nonzero identity and
all modules are considered to be unitary. Prime submodules play an important
role in the module theory over commutative rings. Let M be a module over a
commutative ring R. A prime (resp. primary) submodule N of M is a proper
submodule N of M with the property that for a ∈ R and m ∈ M, am ∈ N
implies that m ∈ N or a ∈ (N :R M) (resp. a

k ∈ (N :R M) for some positive

integer k). In this case P = (N :R M) (resp. P =
√

(N :R M)) is a prime
ideal of R and we say that N is a P-prime (resp. P-primary) submodule of
M. There are several ways to generalize the notion of prime submodules. We
could restrict where am lies or we can restrict where a and/or b lie. We begin
by mentioning some examples obtained by restricting where ab lies. Weakly
prime submodules were introduced by Ebrahimi Atani and Farzalipour in [8].
A proper submodule N of M is weakly prime if for a ∈ R and m ∈ M with
0 6= am ∈ N, either m ∈ N or a ∈ (N :R M). Behboodi and Koohi in [3]
defined another class of submodules and called it classical prime. A proper
submodule N of M is said to be classical prime when for a, b ∈ R and m ∈ M,
abm ∈ N implies that am ∈ N or bm ∈ N.

Recently, M. Baziar and M. Behboodi [2] defined a classical primary submod-
ule in the R-module M as a proper submodule Q of M such that if abm ∈ Q,
where a, b ∈ R and m ∈ M, then either bm ∈ Q or akm ∈ Q for some positive
integer k. Clearly, in case M = R, classical primary submodules coincide with
primary ideals.

Let G be an arbitrary group. A commutative ring R with a non-zero identity
is G-graded if it has a direct sum decomposition R =

⊕

g∈G
Rg such that for

c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 160



Zariski topology on the spectrum of graded classical prime submodules

all g, h ∈ G, RgRh ⊆ Rgh. The G-graded ring R is called a graded integral
domain provided that ab = 0 implies that either a = 0 or b = 0 where a, b ∈
h(R) :=

⋃

g∈G
Rg. If R is G-graded, then an R-module M is said to be G-

graded if it has a direct sum decomposition M =
⊕

g∈G
Mg such that for all

g, h ∈ G, RgMh ⊆ Mgh . For every g ∈ G, an element of Rg or Mg is said to
be a homogeneous element. We denote by h(M) the set of all homogeneous
elements of M, that is h(M) =

⋃

g∈G
Mg. Let M be a G-graded R-module. A

submodule N of M is called graded (or homogeneous) if N =
⊕

g∈G
(N ∩ Mg)

or equivalently N is generated by homogeneous elements. Moreover, M/N
becomes a G-graded R-module with g-component (M/N)g = (Mg + N)/N for
each g ∈ G. An ideal I of R is called a graded ideal if it is a graded submodule
of R and a graded R-module.

Let R be a G-graded ring. A proper graded ideal I of R is said to be
a graded prime ideal if whenever ab ∈ I, we have a ∈ I or b ∈ I, where
a, b ∈ h(R). The graded radical of I , denoted by Gr(I), is the set of all
x ∈ R such that for each g ∈ G there exists ng > 0 with x

ng ∈ I. A graded
R-module M is called graded finitely generated if M =

∑n

i=1 Rxgi, where
xgi ∈ h(M) for every 1 ≤ i ≤ n. It is clear that a graded module is finitely
generated if and only if it is graded finitely generated. For M, consider the
subset T g(M) = {m ∈ M : rm = 0 for some nonzero r ∈ h(R)}. If R is a
graded integral domain, then T g(M) is a graded submodule of M. M is called
graded torsion-free (g-torsion-free for short) if T g(M) = 0, and it is called
graded torsion (g-torsion for short) if T g(M) = M. It is clear that if M is
torsion-free, then it is g-torsion-free. Moreover, if M is g-torsion, then it is
torsion.

Let R be a G-graded ring and M a graded R-module. We recall from [8]
that a proper graded submodule N of M is called graded prime (resp. graded
primary) if rm ∈ N, then m ∈ N or r ∈ (N :R M) = {r ∈ R|rM ⊆ N} (resp.
rk ∈ (N :R M) for some positive integer k), where r ∈ h(R), m ∈ h(M). It is
shown in [8, Proposition 2.7] that if N is a graded prime submodule of M, then
P := (N :R M) is a graded prime ideal of R, and N is called graded P-prime
submodule. Let N be a graded submodule of M. Then N is a graded prime
submodule of M if and only if P := (N :R M) is a graded prime ideal of R and
M/N is a g-torsion-free R/P-module. Note that some graded R-modules M
have no graded prime submodules. We call such graded modules g-primeless.
A submodule S of M will be called graded semiprime if S is an intersection
of graded prime submodules of M. Let Specg(M) denote the set of all graded
prime submodules of M. Let N be a graded submodule of M. The graded
radical of N in M, denoted by GrM (N) is defined to be the intersection of all
graded prime submodules of M containing N [10]. Hence GrM (N) is a graded
semiprime submodule. A proper graded submodule N of M is called graded
weakly prime if 0 6= rm ∈ N, then m ∈ N or r ∈ (N :R M). Hence every
graded prime submodule is graded weakly prime.

c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 161



A. Yousefian Darani and S. Motmaen

From now on, R is a G-graded ring and M is a graded R-module unless
otherwise stated.

3. Graded classical prime submodules

A proper graded submodule N of M is called graded classical prime if for
every a, b ∈ h(R), m ∈ h(M), whenever abm ∈ N, then either am ∈ N
or bm ∈ N. Let N be a graded classical prime submodule of M. Then, it
is easy to see that Ng is a classical prime submodule of the Re-module Mg
for every g ∈ G. It is evident that every graded prime submodule is graded
classical prime. However the next example shows that a graded classical prime
submodule is not necessarily graded prime.

Example 3.1. Assume that R is a graded integral domain and P is a non-zero
graded prime ideal of R. In this case the ideal Q := P ⊕ 0 is a graded classical
prime submodule of the graded R-module R ⊕ R while it is not graded prime.
This example shows also that a graded classical prime submodule need not be
classical prime.

We denote by Cl.Specg(M), the set of all graded classical prime submodules
of M. Obviously, some graded R-modules M have no graded classical prime
submodules; such modules are called g-Cl.primeless. For example, the zero
module is clearly g-Cl.primeless. A submodule S of M will be called graded
classical semiprime if S is an intersection of graded classical prime submodules
of M. Let N be a graded submodule of M. The graded classical radical of N in
M, denoted by Cl.GrM (N), is defined to be the intersection of M and all graded
classical prime submodules of M containing N. So if Cl.Specg(M) = ∅, then
GrclM (N) = M, and if Cl.Specg(M) 6= ∅, then Gr

cl
M (N) is a graded classical

semiprime submodule. If N = 0, then GrclM (0) is called the graded classical
nil-radical of M.

We know that Specg(M) ⊆ Cl.Specg(M). As it is mentioned in example
3.1, it happens sometimes that this containment is strict. We call M a graded
compatible R-module if its graded classical prime submodules and graded prime
submodules coincide, that is if Specg(M) = Cl.Specg(M). If R is a G-graded
ring, then every graded classical prime ideal of R is a graded prime ideal. So,
if we consider R as a graded R-module, it is graded compatible.

The following lemma is obvious.

Lemma 3.2. Let N be a proper graded submodule of M. Then N is a graded
classical prime submodule if and only if for each x ∈ h(M) \ N, (N :R x) is a
graded prime ideal of R.

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Zariski topology on the spectrum of graded classical prime submodules

Proposition 3.3.

(1) Let N be proper graded submodule of M. Then N is a graded prime
submodule of M if and only if N is graded primary and graded classical
prime.

(2) Assume that N and K are graded submodule of M with K ⊆ N. Then
N is a graded classical prime submodule of M if and only if N/K is a
graded classical prime submodule of the graded R-module M/K.

Proof. (1) If N is a graded prime submodule of M, then it clearly is both graded
primary and graded classical prime. Now assume that N is a graded primary
and graded classical prime submodule of M. Let am ∈ N but m /∈ N, where
a ∈ h(R) and m ∈ h(M). Since N is graded primary, there exists a positive
integer k such that ak ∈ (N :R M). Therefore, for every y ∈ h(M) \ N,
ak ∈ (N :R y) and (N :R y) is a prime ideal of R by Lemma 3.2. Hence
a ∈ (N :R y). It follows that a ∈ (N :R M), i.e. N is graded prime in M.

(2) Straightforward. �

Let R be a G-graded R-module and consider Specg(R), the spectrum of
all graded prime ideals of R. The Zariski topology on Specg(R) is defined
in a similar way to that of Spec(R). For each graded ideal I of R, the
graded variety of I is the set V

g
R
(I) = {P ∈ Specg(R)|I ⊆ P}. Then the

set {V
g
R
(I)|I is a graded ideal of R} satisfies the axioms for the closed sets of

a topology on Specg(R), called the Zariski topology on Specg(R) (see [14]).
In [16], Specg(M) has endowed with quasi-Zariski topology. For each graded

submodule N of M, let V
g
∗ (N) = {P ∈ Specg(M)|N ⊆ P}. In this case, the

set ζ
g
∗ (M) = {V

g
∗ (N)|N is a graded submodule of M} contains the empty set

and Specg(M), and it is closed under arbitrary intersections, but it is not
necessarily closed under finite unions. The graded R-module M is said to be
a g-Top module if ζ

g
∗ (M) is closed under finite unions. In this case ζ

g
∗ (M)

satisfies the axioms for the closed sets of a unique topology τ
g
∗ on Specg(M).

The topology τ
g
∗ (M) on Specg(M) is called the quasi-Zariski topology. In the

remainder of this section we use a similar method to define a topology on
Cl.Specg(M). To this end, For each graded submodule N of M, set

Vg∗(N) = {P ∈ Cl.Specg(M)|N ⊆ P}.

Proposition 3.4. Let M be a graded R-module. Then

(1) For each subset E ⊆ h(M), Vg∗(E) = V
g
∗(N) = V

g
∗ (Gr

cl
M (N)), where N

is the graded submodule of M generated by E.
(2) Vg∗(0) = Cl.Specg(M), and V

g
∗(M) = ∅.

(3) If {Nλ}λ∈Λ is a family of graded submodules of M, then
⋂

λ∈Λ V
g
∗(Nλ) =

Vg∗(
∑

λ∈Λ Nλ).
(4) For every pair N and K of graded submodules of M, we have Vg∗(N) ∪

Vg∗(K) ⊆ V
g
∗(N ∩ K).

Proof. The proof of (2) − (4) is easy. So we just provide a proof for part (1).
Assume that N is the graded submodule of M generated by E ⊆ h(M). Then
from E ⊆ N ⊆ GrclM (N) we have

c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 163



A. Yousefian Darani and S. Motmaen

Vg∗(Gr
cl
M (N)) ⊆ V

g
∗(N) ⊆ V

g
∗(E).

On the other hand, N is the smallest graded submodule of M containing E,
so that if P ∈ Vg∗(E), then P ∈ V

g
∗(N). Therefore V

g
∗(E) = V

g
∗(N). More-

over GrclM (N) is the intersection of all graded classical prime submodules of
M containing N; so Vg∗(N) = V

g
∗(Gr

cl
M (N)). Therefore V

g
∗(E) = V

g
∗(N) =

Vg∗(Gr
cl
M (N)). �

Now if we set

η
g
∗(M) = {V

g
∗(N)|N is a graded submodule of M}

then η
g
∗(M) contains the empty set and Cl.Specg(M). Moreover η

g
∗(M) is

closed under arbitrary intersections, but it is not necessarily closed under finite
unions.

Definition 3.5. Let M be a graded R-module.

(1) We shall say that M is a g-Cl.Top module if η
g
∗(M) is closed under

finite unions, i.e. for any graded submodules N and L of M there
exists a graded submodule K of M such that Vg∗(N) ∪ V

g
∗(L) = V

g
∗(K).

(2) A graded classical prime submodule N of M will be called graded clas-
sical extraordinary, or g-Cl.extraordinary for short, if whenever K and
L are graded classical semiprime submodules of M with K ∩ L ⊆ N
then K ⊆ N or L ⊆ N.

Note that if M is a g-Cl.Top module, then η
g
∗(M) satisfies the axioms for the

closed sets of a unique topology ̺
g
∗ on Cl.Specg(M). In this case, the topology

̺
g
∗(M) on Cl.Specg(M) is called the quasi-Zariski topology. Note that we are

not excluding the trivial case where Cl.Specg(M) is empty; that is every g-
Cl.primeless modules is a g-Cl.Top module. The next result is a useful tool for
characterizing g-Cl.Top modules.

Theorem 3.6. Let M be a graded R-module. Then, the following statements
are equivalent:

(i) M is a g-Cl.Top module.
(ii) Every graded classical prime submodule of M is g-Cl.extraordinary.
(iii) Vg∗(N) ∪ V

g
∗(L) = V

g
∗(N ∩ L) for any graded classical semiprime sub-

modules N and L of M.

Proof. The result is clear when Cl.Specg(M) = ∅. So assume that Cl.Specg(M)
6= ∅.

(i) ⇒ (ii) Let M be a g-Cl.Top module. Assume that P is a graded classical
prime submodule of M and that N, L are graded classical semiprime submod-
ules of M with N ∩L ⊆ P . By assumption, there exists a graded submodule K
of M with Vg∗(N) ∪ V

g
∗(L) = V

g
∗(K). Since N is a graded classical semiprime

submodule, N =
⋂

i∈I
Pi in which {Pi}i∈I is a collection of graded classical

prime submodules of M. For every i ∈ I, we have

Pi ∈ V
g
∗(N) ⊆ V

g
∗(K) ⇒ K ⊆ Pi ⇒ K ⊆

⋂

i∈I
Pi = N

c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 164



Zariski topology on the spectrum of graded classical prime submodules

Similarly, K ⊆ L. So K ⊆ N ∩ L. Now we have

Vg∗(N) ∪ V
g
∗(L) ⊆ V

g
∗(N ∩ L) ⊆ V

g
∗(K) = V

g
∗(N) ∪ V

g
∗(L).

Consequently, Vg∗(N) ∪ V
g
∗(L) = V

g
∗(N ∩ L). Now from N ∩ L ⊆ P we have

P ∈ Vg∗(N ∩L) = V
g
∗(N)∪V

g
∗(L). Hence either P ∈ V

g
∗(N) or P ∈ V

g
∗(L), that

is either N ⊆ P or L ⊆ P . So P is g-Cl.extraordinary.
(ii) ⇒ (iii) Suppose that every graded classical prime submodule of M is

g-Cl.extraordinary. Assume that N and L are two graded classical semiprime
submodules of M. Clearly Vg∗(N) ∪ V

g
∗(L) ⊆ V

g
∗(N ∩ L). Now assume that

P ∈ Vg∗(N∩L). Then N∩L ⊆ P . Since P is g-Cl.extraordinary, we have N ⊆ P
or L ⊆ P , that is either P ∈ Vg∗(N) or P ∈ V

g
∗(L). Therefore V

g
∗(N ∩ L) ⊆

Vg∗(N) ∪ V
g
∗(L), and so V

g
∗(N) ∪ V

g
∗(L) = V

g
∗(N ∩ L).

(iii) ⇒ (i) Let N, L be two graded submodules of M. We can assume that
Vg∗(N) and V

g
∗(L) are both nonempty, for otherwise V

g
∗(N) ∪ V

g
∗(L) = V

g
∗(N)

or Vg∗(N) ∪ V
g
∗(L) = V

g
∗(L). We know that Gr

cl
M (N) and Gr

cl
M (L) are both

graded classical semiprime submodules of M. Setting K = GrclM (N) ∩ Gr
cl
M (L)

we have:

Vg∗(N) ∪ V
g
∗(L) = V

g
∗(Gr

cl
M (N)) ∪ V

g
∗(Gr

cl
M (L)) = V

g
∗(Gr

cl
M (N) ∩ Gr

cl
M (L)) =

Vg∗(K)

by (iii). Hence M is a g-Cl.Top module. �

Corollary 3.7. Every g-Cl.Top module is a g-Top module.

Proof. Assume that M is a g-Cl.Top module. Let P be a graded prime sub-
module of M. Since every graded prime L-submodule is a graded classical
prime submodule, P is g-Cl.extraordinary by Proposition 3.6. Hence it is g-
extraordinary. Now the result follows from [16, Theorem 2.3]. �

Theorem 3.8. Let M be a g-Cl.Top R-module. Then,

(1) For every graded submodule K of M, the R-module M/K is a g-Cl.Top
module.

(2) The graded RP -module MP is a g-Cl.Top module for every graded prime
ideal P of R.

(3) If GrclM (N) = N for every graded submodule N of M, then M is a
graded distributive module.

Proof. There will be nothing to prove if M has no graded classical prime sub-
modules. So assume that Cl.Specg(M) 6= ∅.

(1) By Proposition 3.3, the graded classical prime submodules of M/K are
just the submodules N/K where N is a graded classical prime submodule of M
with K ⊆ N. Consequently, any graded classical semiprime submodule of M/K
is of the form S/K in which S is a graded classical semiprime submodule of M
with K ⊆ S. Assume that S1/K and S2/K are two graded classical semiprime
submodules of M/K. Then, by Theorem 3.6, Vg∗(S1) ∪ V

g
∗(S2) = V

g
∗(S1 ∩ S2)

since M is a g − Cl.T op module. Thus Vg∗(S1/K) ∪ V
g
∗(S2/K) = V

g
∗(S1/K ∩

S2/K). It follows from Theorem 3.6 that M/K is a g − Cl.T op module.

c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 165



A. Yousefian Darani and S. Motmaen

(2) By Theorem 3.6, it is enough to show that every graded classical prime
submodule of MP is g-Cl.extraordinary. Let N be a graded classical prime
submodule of MP , and let S1 ∩ S2 ⊆ N for some graded classical semiprime
submodules S1, S2 of MP . Clearly, N ∩M is a proper graded submodule of M.
Assume that a, b ∈ h(R) and m ∈ h(M) are such that abm ∈ N ∩ M. Then,
a/1, b/1 ∈ h(RP ) and m/1 ∈ h(MP ) with (a/1)(b/1)(m/1) = (abm)/1 ∈ N.
It follows that either (a/1)(m/1) ∈ N or (b/1)(m/1) ∈ N since N is graded
classical prime. Therefore, either am ∈ N ∩ M or bm ∈ N ∩ M. This implies
that N ∩ M is a graded classical prime submodule of M. Hence N is g-
Cl.extraordinary by Theorem 3.6. As another consequence, S1 ∩M and S2 ∩M
are graded classical semiprime submodules of M with (S1 ∩ M) ∩ (S2 ∩ M) ⊆
N ∩M. Therefore, S1 ∩M ⊆ N ∩M or S2 ∩M ⊆ N ∩M. It follows that either
S1 = (S1 ∩M)RP ⊆ (N ∩M)RP = N or S2 = (S2 ∩M)RP ⊆ (N ∩M)RP = N.
Hence N is a g-Cl.extraordinary submodule of MP .

(3) For every graded submodules N, K and L of M we have:
(K + L) ∩ N = GrclM ((K + L) ∩ N)

=
⋂

{P |P ∈ Vg∗((K + L) ∩ N)}
=

⋂

{P |P ∈ Vg∗(K + L) ∪ V
g
∗(N)}

=
⋂

{P |P ∈ (Vg∗(K) ∩ V
g
∗(L)) ∪ V

g
∗(N)}

=
⋂

{P |P ∈ (Vg∗(K)∪V
g
∗(N))∩(V

g
∗(L)∪V

g
∗(N))}

=
⋂

{P |P ∈ (Vg∗(K ∩ N)) ∩ (V
g
∗(L ∩ N))}

=
⋂

{P |P ∈ Vg∗((K ∩ N) + (L ∩ N))}
= GrclM ((K ∩N)+(L∩N)) = (K ∩N)+(L∩N).

Thus M is graded distributive. �

Let M be a g-Cl.Top module and let X = Cl.Specg(M). We know that any
closed subset of X is of the form Vg∗(N) for some graded submodule N of M.
But now the question arises as to what open subsets of X look like. To say
that any open subset of X is of the form X − Vg∗(N) for some graded prime
submodule N of M, though true, is not very helpful. For every subset S of
h(M), define

XS = X − V
g
∗(S)

In particular, if S = {f}, we denote XS be Xf .

Proposition 3.9. The set {Xf|f ∈ h(M)} is a basis for the quasi-Zariski
topology on X.

Proof. Let U be a non-void open subset in X. Then U = X − Vg∗(N) for
some graded submodule N of M. Assume that N is generated by some subset
E ⊆ h(M). Then we have

U = X − Vg∗(N) = X − V
g
∗(E) = X − V

g
∗(
⋃

f∈E
{f}) = X −

⋂

f∈E
Vg∗(f) =

⋃

f∈E
(X − Vg∗(f)) =

⋃

f∈E
Xf

Therefore the set {Xf |f ∈ h(M)} is a basis for X. �

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Zariski topology on the spectrum of graded classical prime submodules

A topological space X is said to be irreducible if X 6= ∅ and if every pair
of non-void open sets in X intersect. Let X be a topological space. A subset
A ⊆ X is said to be dense in X if and only if A ∩ G 6= ∅ for every non-void
open subset G ⊆ X. Therefore X is irreducible if and only if every non-void
open subset of X is dense.

Lemma 3.10. Let M be a graded R-module. Then, N := GrclM (0) is a graded
classical prime submodule of M if and only if Cl.Specg(M) is irreducible.

Proof. Set X = Cl.Specg(M). Assume first that N is a graded classical prime
submodule of M. Let U, V ⊆ X be non-void open subsets. Pick P ∈ U. Now,
U = X \ Vg∗(E) for some E ⊆ h(M). Then P ∈ U implies that E * P .
Moreover, from N ⊆ P we have E * N, so that N ∈ U. Similarly, N ∈ V .
Hence N ∈ U ∩V , and thus U ∩V 6= ∅. Therefore, X is irreducible. Conversely,
assume that N is not a graded classical prime submodule of M. So there exist
a, b ∈ h(R) and m ∈ h(M) such that am, bm /∈ N, but abm ∈ N. Both Xam
and Xbm are open in X. Also am /∈ N ⇒ V

g
∗(am) 6= X so Xam 6= ∅. Similarly,

Xbm 6= ∅. Now we have,

Xam ∩ Xbm = Xabm = X − V
g
∗
(abm) ⊆ X − Vg

∗
(N) = ∅

Therefore, X is not irreducible. �

4. Homomorphisms and graded classical prime spectrum of
modules

In our discussion so far we have concerned ourselves with the graded classical
prime spectrum of but one graded module at any given time. A natural question
to ask is what relationships on their respective graded classical prime spectra
are induced by a homomorphism between two rings. In this section, we address
this question.

Let M and M′ be two graded R-modules and let φ : M → M′ be a graded
R-homomorphism. The inverse image of a graded classical prime submodule of
M′ is a graded classical prime submodule of M. For every graded submodule
of M′, we write φ−1(N′) = N′

c

, the contraction of N′ to M. Also, if N is a
graded submodule of M, then we write Rφ(N) = Ne, the graded submodule of
M′ generated by φ(N), the extension of N to M′. Let X = Cl.Specg(M) and
Y = Cl.Specg(M

′). Thus if P ∈ Y , then P c ∈ X. So we see that φ induces a
map φ∗ : Y → X defined by φ∗(P) = P c, for all P ∈ Y . Before continuing, we
introduce a more explicit notation:

P ∈ Vg
∗M

(E) means that E ⊆ h(M) and E ⊆ P ∈ X

and

Q ∈ Vg
∗M′

(F) means that F ⊆ h(M′) and F ⊆ P ∈ Y

c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 167



A. Yousefian Darani and S. Motmaen

Proposition 4.1. Let M and M′ be two graded R-modules and let φ : M → M′

be a graded R-homomorphism. Let X = Cl.Specg(M), Y = Cl.Specg(M
′), and

let φ∗ : Y → X be the induced map.

(1) φ∗ is continuous.

(2) If N is a graded submodule of M, then φ∗
−1

(Vg
∗M

(N)) = Vg
∗M′

(Ne).
(3) If φ is an epimorphism, then φ∗ is a homeomorphism from Y onto the

closed subset Vg
∗M

(Ker(φ)) of X.

Proof. (1) It is enough to show that if U is open in X, then φ∗
−1

(U) is open if
Y . For every subset E ⊆ h(M) and Q ∈ Y , we have

φ(E) ⊆ Q ⇔ E ⊆ φ−1(Q)
⇔ φ∗(Q) ∈ Vg

∗M
(E)

⇔ Q ∈ φ∗
−1

(Vg
∗M

(E))

Hence, if f ∈ h(M) and Q ∈ Y , then

Q ∈ Yφ(f) ⇔ φ(f) /∈ Q

⇔ Q /∈ φ∗
−1

(Vg
∗M

(f))

⇔ Q ∈ φ∗
−1

(X) − φ∗
−1

(Vg
∗M

(f))

⇔ Q ∈ φ∗
−1

(X − Vg
∗M

(f)) = φ∗
−1

(Xf )

Therefore, φ∗
−1

(Xf ) = Yφ(f). In particular, if U is open in X, then φ
∗
−1

(U)
is open if Y . Hence φ∗ is continuous.

(2) Assume that Q ∈ Y . Then,

Q ∈ φ∗
−1

(Vg
∗M

(N)) ⇔ φ(N) ⊆ Q
⇔ Q ∈ Vg

∗M′
(φ(N))

⇔ Q ∈ Vg
∗M′

(Ne)

Therefore, φ∗
−1

(Vg
∗M

(N)) = Vg
∗M′

(Ne).
(3) Suppose that φ is an epimorphism. Then, there exists a one-to-one cor-

respondence between graded submodules of M′ and graded submodules of M
containing Ker(φ). Under this correspondence, graded classical prime submod-
ules of M′ correspond to the graded classical prime submodules of M containing
Ker(φ). Therefore, φ∗ : Y → Vg

∗M
(Ker(φ)) is bijective. As φ∗ is continuous

by (1), it suffices to prove that φ∗ is an open map. Assume that U is an open
subset of Y . Then, without loss of generality, we may assume that U = Yf for
some f ∈ h(M′). In this case,

φ∗(U) = φ∗(Yf ) = {φ
∗(Q)|Q ∈ Y and f /∈ Q}

= {P ∈ Vg
∗M

(Ker(φ))|φ−1(f) /∈ P}
= Xφ−1(f) ∩ V

g
∗M

(Ker(φ))

This implies that φ∗(U) is an open subset of Vg
∗M

(Ker(φ)), that is φ∗ is an
open map. Consequently, φ∗ : Y → Vg

∗M
(Ker(φ)) is a homeomorphism. �

Corollary 4.2. Let M be a graded R-module, and let N be the graded clas-
sical nil-radical of M. Then Cl.Specg(M) and Cl.Specg(M/N) are naturally
homeomorphic.

c© AGT, UPV, 2013 Appl. Gen. Topol. 14, no. 2 168



Zariski topology on the spectrum of graded classical prime submodules

Proof. By Proposition 4.1, the canonical graded R-epimorphism f : M → M/N
induces the homeomorphism f∗ : Cl.Specg(M/N) → V

g
∗M

(Ker(f)). Now the
result follows from Vg

∗M
(Ker(f)) = Vg

∗M
(N) = Cl.Specg(M). �

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