@ Appl. Gen. Topol. 22, no. 2 (2021), 251-257doi:10.4995/agt.2021.13225
© AGT, UPV, 2021

A new topology over the primary-like spectrum

of a module

Fatemeh Rashedi

Department of Mathematics, Velayat University, Iranshahr, Iran (f.rashedi@velayat.ac.ir)

Communicated by J. Galindo

Abstract

Let R be a commutative ring with identity and M a unitary R-module.
The primary-like spectrum SpecL(M) is the collection of all primary-
like submodules Q of M, the recent generalization of primary ideals,
such that M/Q is a primeful R-module. In this article, we topolo-
gies SpecL(M) with the patch-like topology, and show that when,
SpecL(M) with the patch-like topology is a quasi-compact, Hausdorff,
totally disconnected space.

2010 MSC: 13C13; 13C99.

Keywords: primary-like submodule; Zariski topology; patch-like topology.

1. Introduction

Throughout this paper all rings are commutative with identity and all mod-
ules are unitary. For a submodule N of M, we let (N : M) denote the ideal
{r ∈ R | rM ⊆ N} and annihilator of M, denoted by Ann(M), is the ideal
(0 : M). By a prime submodule (or a p-prime submodule) of M, we mean
a proper submodule P with p = (P : M) such that rm ∈ P for r ∈ R and
m ∈ M implies that either m ∈ P or r ∈ p. The prime spectrum (or simply,
the spectrum) of M, denoted by Spec(M), is the set of all prime submodules of
M [1, 2, 3, 5]. The intersection of all prime submodules of M containing N is
called the radical of N and denoted by radN. If there is no prime submodule
containing N, then we define radN = M. As a new generalization of a primary
ideal on the one hand and a generalization of a prime submodule on the other

Received 04 March 2020 – Accepted 24 May 2021

http://dx.doi.org/10.4995/agt.2021.13225


F. Rashedi

hand, a proper submodule Q of M is said to be primary-like if rm ∈ Q implies
r ∈ (Q : M) or m ∈ radQ ([6]). We say that a submodule N of an R-module M
satisfies the primeful property if for each prime ideal p of R with (N : M) ⊆ p,
there exists a prime submodule P containing N such that (P : M) = p. If the
zero submodule of M satisfies the primeful property, then M is called primeful.
For instance finitely generated modules, projective modules over domains and
(finite and infinite dimensional) vector spaces are primeful (see [9]). It is easy
to see that, if Q is a primary-like submodule satisfying the primeful property,
then p =

√

(Q : M) is a prime ideal of R and so in this case, Q is called a p-
primary-like submodule. The primary-like spectrum SpecL(M) is defined to be
the set of all primary-like submodules of M satisfying the primeful property. If
Q ∈ SpecL(M), since Q satisfies the primeful property, there exists a maximal
ideal m of R and a prime submodule P containing Q such that (P : M) = m
and so radQ 6= M.

For any submodule N of M, let

ν(N) = {Q ∈ SpecL(M)|
√

(Q : M) ⊇
√

(N : M)}.

Then we have the following lemma.

Lemma 1.1. Let M be an R-module. Let N, N′ and {Ni|i ∈ I} be submodules
of M. Then the following hold.

(1) ν(M) = ∅.
(2) ν(0) = SpecL(M).
(3) If N ⊆ N′, then ν(N′) ⊆ ν(N).
(4)

⋂

i∈I
ν(Ni) = ν(

∑

i∈I
(Ni : M)M).

(5) ν(N) ∪ ν(N′) = ν(N ∩ N′).
(6) ν(radN) ⊆ ν(N).

(7) If
√

(N : M) =
√

(N′ : M), then ν(N) = ν(N′). The converse is also
true if both N and N′ are primary-like.

(8) ν(N) = ν(
√

(N : M)M).

Also, for each submodule N of M we denote the complement of ν(N) in
SpecL(M) by U(N). From (1), (2), (4) and (5) above, the family η(M) =
{U(N)|N ≤ M} is closed under finite intersections and arbitrary unions. More-
over, we have U(M) = SpecL(M) and U(0) = ∅. Therefore, η(M), as the
family of all open sets, satisfy the axioms of a topology T on SpecL(M), called
the Zariski topology on M.
In Section 2, we topologies SpecL(M) with a patch-like topology, and show
that, if M is a Noetherian multiplication R-module and (N : M) is a radical
ideal for every submodule N of M, then SpecL(M) with the patch-like topology
is a quasi-compact, Hausdorff, totally disconnected space (Corollary 2.16).

2. Main Results

We need to recall the patch topology (see [7, 8], for definition and more
details). Let X be topological space. By the patch topology on X, we mean
the topology which has as a sub-basis for its closed sets the closed sets and

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A new topology over the primary-like spectrum

compact open sets of the original space. By a patch we mean a set closed
in the patch topology. The patch topology associated to a spectral space is
compact and Hausdorff (see [8]). Also, the patch topology associated to the
Zariski topology of a ring R (not necessarily commutative) with ACC on ideals
is compact and Hausdorff (see [7, Proposition 16.1]).

Definition 2.1. Let M be an R-module, and let ω(M) be the family of all
subsets of SpecL(M) of the form ν(N)∪U(K) where ν(N) is any Zariski-closed
subset of SpecL(M) and U(K) is a Zariski-quasi-compact subset of SpecL(M).
Clearly ω(M) is closed under finite unions and contains SpecL(M) and the
empty set, since SpecL(M) equals ν(0)∪U(0) and the empty set equals ν(M)∪
U(0). Therefore ω(M) is basis for the family of closed sets of a topology on
SpecL(M), and call it patch-like topology of M. Thus

ω(M) = {ν(N) ∪ U(K)|N, K ≤ M, U(K) is Zariski-quasi-compact},

and hence we obtain the family

Ω(M) = {ν(N) ∩ U(K)|N, K ≤ M, U(K) is Zariski-quasi-compact},

which is a basis for the open sets of the patch-like topology, i.e., the patch-
like-open subsets of SpecL(M) are precisely the unions of sets from Ω(M). We
denote the patch-like topology of SpecL(M) by Tp(M).

Definition 2.2. Let M be an R-module, and let Ω̃(M) be the family of all

subsets of SpecL(M) of the form ν(N)∩U(K) where N, K ≤ M. Clearly Ω̃(M)
contains SpecL(M) and the empty set, since SpecL(M) equals ν(0)∩U(M) and

the empty set equals ν(M)∩U(0). Let T̃p(M) to be the collection Ũ of all unions

of elements of Ω̃(M). Then T̃p(M) is a topology on SpecL(M) and it is called

the finer patch-like topology (in fact, Ω̃(M) is a basis for the finer patch-like
topology of M).

We will use X to represent SpecL(M).

Lemma 2.3. Let M be an R-module and Q ∈ X. Then for each finer
patch-like-neighborhood W of Q, there exists a submodule L of M such that
√

(Q : M) ⊆
√

(L : M) and Q ∈ ν(Q) ∩ U(L) ⊆ W.

Proof. Since Q ∈ W, there exists a neighborhood of the form ν(K)∩U(N) ⊆ W

such that Q ∈ ν(K) ∩ U(N) where
√

(Q : M) ⊇
√

(K : M) and
√

(Q : M) +
√

(N : M). Since Q ∈ ν(Q) and ν(Q) ⊆ ν(K), we may replace ν(K) by ν(Q).

Now we claim that ν(Q) ∩ U(N) = ν(Q) ∩ U((I + p)M), where p =
√

(Q : M)

and I =
√

(N : M). Since U(IM) ⊆ U((I + p)M),

ν(Q) ∩ U(N) = ν(Q) ∩ U(IM) ⊆ ν(Q) ∩ U((I + p)M).

Suppose that Q′ ∈ ν(Q) ∩ U((I + p)M), then Q′ /∈ U(Q). On the other
hand Q′ ∈ U((I + p)M) = U(N) ∪ U(Q). This follows that Q′ ∈ U(N).
Thus ν(Q) ∩ U(N) = ν(Q) ∩ U((I + p)M). Now let L = (I + p)M. Then

p ⊆ I + p ⊆
√

(L : M) and Q ∈ ν(Q) ∩ U(L) ⊆ W. �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 253



F. Rashedi

Let Y be a subset of Y for a module M. We will denote the closure of Y in
X with finer patch-like topology by Y.

Proposition 2.4. Let M be an R-module and Y ⊆ X be a finite set. If
Q ∈ Y with finer patch-like topology, then there exists A ⊆ Y such that ν(Q) =
ν(
⋂

Q′∈A
Q′).

Proof. Suppose Q ∈ Y. If Q ∈ Y , then we are thorough. Thus we can
assume that Q /∈ Y. Let A = {Q′ ∈ Y|

√

(Q : M) ⊂
√

(Q′ : M)}. Since Q ∈
U(M) ∩ ν(Q), there exists Q′′ ∈ Y such that Q′′ ∈ U(M) ∩ ν(Q). Since Q /∈ Y,
√

(Q : M) ⊂
√

(Q′′ : M) and hence A 6= ∅. Since
√

(Q : M) ⊂
√

(Q′ : M) for
each Q′ ∈ A,

√

(Q : M) ⊂ ∩Q′∈A
√

(Q′ : M) =
√

(∩Q′∈AQ′ : M).

If ∩Q′∈A
√

(Q′ : M) *
√

(Q : M), then Q ∈ U(∩Q′∈AQ
′) ∩ ν(Q). Since Q ∈ Y,

there exists Q′′ ∈ Y such that Q′′ ∈ U(∩Q′∈AQ
′) ∩ ν(Q). Therefore Q′′ ∈ ν(Q)

and hence Q′′ ∈ A. But
⋂

Q′∈A

√

(Q′ : M) =
√

(
⋂

Q′∈A
Q′ : M) ⊆

√

(Q′′ : M).

Thus Q′′ /∈ U(
⋂

Q′∈A
Q′), a contradiction. Thus

⋂

Q′∈A

√

(Q′ : M) ⊆
√

(Q : M),
and hence

ν(Q) = ν(
⋂

Q′∈A
Q′).

�

A module M over a commutative ring R is called a multiplication module if
each submodule of M has the form IM for some ideal I of R [4]. In this case
we can take I = (N : M).

Proposition 2.5. Let M be a multiplication R-module such that (Q : M) is
a radical ideal for every Q ∈ X. Then X with the finer patch-like topology is
Hausdorff. Moreover, X with this topology is totally disconnected.

Proof. Assume Q, Q′ ∈ X are distinct points. Since Q 6= Q′, (Q : M) 6= (Q′ :
M). Thus either (Q : M) * (Q′ : M) or (Q′ : M) * (Q : M). Suppose that
(Q : M) * (Q′ : M). By Definition 2.2, U1 := U(M) ∩ ν(Q) is a finer patch-
like-neighborhood of Q and since (Q : M) * (Q′ : M), U2 := U(Q) ∩ ν(Q

′)
is a finer patch-like-neighborhood of Q′. Clearly U(Q) ∩ ν(Q) = ∅ and hence
U1 ∩ U2 = ∅. Thus X is a Hausdorff space. On the other hand for every
submodule N of M, observer that the sets U(N) and ν(N) are open in finer
patch-like topology, since ν(N) = U(M)∩ν(N) and U(N) = U(N)∩ν(0). Since
U(N) and ν(N) are complement of each other, they are both finer both-closed
as well. Therefore the finer patch-like topology on X has a basis of open sets
which are also closed, and hence X is totally disconnected in this topology. �

The following example shows that the condition multiplication in Proposi-
tion 2.5 is necessary.

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A new topology over the primary-like spectrum

Example 2.6. Let V be a vector space over a field F with dimF V > 1. It is
evident that X and Spec(V ) are the set of all proper vector subspaces of V .

Now,
√

(Q : M) =
√

(Q′ : M) for all distinct subspaces Q, Q′ ∈ X . If (Q : M)
is a radical ideal for every Q ∈ X , then X with the finer patch-like topology is
not Hausdorff.

Definition 2.7. An R-module M is called p-module if for each prime ideal p
of R such that (pM : M) = p, there exists Q ∈ X such that

√

(Q : M) = p.

For example every finitely generated faithful module is a p-module. Now we
show that every Noetherian R-module M is also a p-module.
Let p be a prime ideal of a ring R, M an R-module, and N ≤ M. By the
saturation of N with respect to p, we mean the contraction of Np in M and
designate it by Sp(N). It is also known that Sp(N) = {m ∈ M|rm ∈ N for
some r ∈ R\p}. Saturations of submodules were investigated in detail in [10].

Lemma 2.8. Let M be a Notherian R-module. Then M is a p-module.

Proof. Assume M is a Notherian R-module. Hence M is finitely generated. By
[11, Proposition 1.8], for each prime ideal p of R, Sp(pM) is a prime submodule
of M such that (pM : M) = p. Thus Sp(pM) ∈ X . �

Theorem 2.9. Let R be a ring and M be a p-module such that R/Ann(M)
has ACC on ideals. If (N : M) is a radical ideal for every submodule N of M,
then X with the finer patch-like topology is a quasi-compact space.

Proof. Suppose M is a p-module and R/Ann(M) has ACC on ideals. Assume
A is a family of finer patch-like-open sets covering X and suppose that no finite
subfamily of A covers X . Suppose

S = {I|I is an ideal of R such that Ann(M) ⊆ I and no finite subfamily of A
covers ν(IM)}.

Since ν(Ann(M)M) = ν(0) = X , S 6= ∅. We may use the ACC on ideals of
R/Ann(M) to choose an ideal m of R maximal with respect to the property
that no finite subfamily of A covers ν(mM) (i.e., m is a maximal element of
S). It is clear that mM 6= M. We claim that m is a prime ideal of R, for
if not, suppose that I and J are two ideals of R properly containing m and
IJ ⊆ m. Then ν(IM) and ν(JM) covered by finite subfamily of A. Suppose

Q ∈ ν(IJM), then IJ ⊆ p :=
√

(Q : M). Since p is prime, either I ⊆ p or
J ⊆ p, and hence either Q ∈ ν(IM) or Q ∈ ν(JM). Thus ν(IJM) covered by
a finite subfamily of A. Since IJ ⊆ m, then ν(mM) ⊆ ν(IJM). Thus ν(mM)
covered by finite subfamily of A, a contradiction. Thus m is a prime ideal of
R. We claim that (mM : M) = m, for if not, then there exists an ideal m1 of
R such that m1 = (mM : M) and m ⊂ m1. This follows that mM = m1M
and so no finite subfamily of A covers ν(m1M), contrary to maximality of m.
Therefore (mM : M) = m and since M is p-module, there exists Q′ ∈ X

such that
√

(Q′ : M) = m. Let U ∈ A such that Q′ ∈ U. By Lemma 2.3,

there exists a submodule K of M such that m =
√

(Q′ : M) ⊆
√

(K : M) and

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 255



F. Rashedi

Q′ ∈ U(K) ∩ ν(Q′) ⊆ U. Suppose (K : M) = I. By Lemma 1.1, we know
that U(K) = U(IM) and ν(Q′) = ν(mM), and so Q′ ∈ U(IM) ∩ ν(mM) ⊆ U.
Since m ⊆ I, then ν(IM) can be covered by some finite subfamily A′ of A. But
ν(mM)\ν(IM) = ν(mM)\[U(IM)]c = ν(mM) ∩ U(IM) ⊆ U and so ν(mM)
can be covered by A′ ∪ {U}, contrary to our choice of Q′. Thus there must
exist a finite subfamily of A which covers X . Therefore X is quasi-compact in
the finer patch-like topology of M. �

It is well-known that if M is a Noetherian module over a ring R, then
R/Ann(M) is a Noetherian ring. Hence we have the following result.

Corollary 2.10. Let M be a Noetherian R-module. If (N : M) is a radical
ideal for every submodule N of M, then X with the finer patch-like topology is
a quasi-compact space.

Proof. Using Lemma 2.8 and Theorem 2.9. �

We need the following evident lemma.

Lemma 2.11. Let T , T ∗ be two topology on X such that T ⊆ T ∗. If X is
quasi-compact in T ∗, then X is also quasi-compact in T .

Theorem 2.12. Let M be an R-module. If X is quasi-compact with the finer
patch-like topology, then for each submodule N of M, U(N) is a quasi-compact
subset of X with the Zariski topology. Consequently, X with the Zariski topology
is quasi-compact.

Proof. By Definition 2.2, for each submodule N of M, ν(N) = ν(N) ∩ U(M)
is an open subset of X with finer patch-like topology, and hence, for each
submodule N of M, U(N) is a closed subset in X with finer patch-like topology.
Since every closed subset of a quasi-compact space is quasi-compact, U(N) is
quasi-compact in X with finer patch-like topology and so by Lemma 2.11, it
is quasi-compact in X with the Zariski topology. Now, since X = U(M), X is
quasi-compact with the Zariski topology. �

Corollary 2.13. Let M be an R-module. If X is quasi-compact with finer
patch-like topology, then the finer patch-like topology and the patch-like topology

of M coincide.

Proof. By Theorem 2.12, for each submodule K of M, U(K) is quasi-compact.
Therefore for each N, K ≤ M, ν(N) ∩ U(K) is an element of the basis Ω(M)
of the patch-like topology on X . �

Corollary 2.14. Let M be an R-module such that (N : M) is a radical ideal
for every submodule N of M. If M is Noetherian or M is a p-module such
that R/Ann(M) has ACC on ideals, then the finer patch-like topology and the
patch-like topology of M coincide.

Proof. By Theorem 2.9 and Corollaries 2.10 and 2.13. �

We conclude this section with the following results.

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A new topology over the primary-like spectrum

Corollary 2.15. Let M be a multiplication p-module such that (N : M) is a
radical ideal for every submodule N of M and R/Ann(M) has ACC on ideals.
Then X with the Zariski topology is a Hausdorf, quasi-compact, totally discon-
nected space.

Proof. By Proposition 2.5, Theorem 2.9 and Corollary 2.13. �

Corollary 2.16. Let M be a multiplication Noetherian R-module such that
(N : M) is a radical ideal for every submodule N of M. Then X with the
Zariski topology is a Hausdorf, quasi-compact, totally disconnected space.

Proof. By Lemma 2.8, Proposition 2.5, Theorem 2.9 and Corollary 2.13. �

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