The Spectrum of the Zariski Topology for Multiplication Krasner Hypermodules

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The Spectrum of the Zariski Topology for Multiplication Krasner Hypermodules

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A peer-reviewed article of this preprint also exists.

Ergül Türkmen^*,

Burcu Nisancı Türkmen

,Öznur KULAK

Ergül Türkmen^*,

Burcu Nisancı Türkmen

,Öznur KULAK

This version is not peer-reviewed

Submitted:

21 January 2023

Posted:

23 January 2023

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Abstract

In this study, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings in [17]. Firstly we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hyper-modules, then we give some relevant properties of the hypermodule in this topological hyperspace.

Keywords:

Subject: Computer Science and Mathematics - Analysis

1. Introduction

Hypergroup theory, which has been defined in [19] as a more comprehensive algebraic structure of group theory and has been investigated by different authors in modern algebra. It has been developed using hyperring and hypermodule theory studies by most authors (see [1,2,3,6,7,8,9,10,15,16,19,20,21,22,26]).

Let’s start by giving the basic information necessary for the algebraic structure that we will study as Krasner S-hypermodule in studying the S-hypermodule class on a fixed Krasner hyperring class S. Let N be a non-empty set.

(N, \cdot)

is called ahypergroupoid if for the map defined as

\cdot : \underset{(x, y)}{N \times N} \underset{\to}{\to} \underset{x \cdot y}{P^{*} (N)}

is a function. Here "·" is called a hyperproduct or hyperoperation on N. Let X and Y be subsets of N. The hyperproduct

X \cdot Y

is defined as

X \cdot Y = \underset{(x, y) \in X \times Y}{\cup} x \cdot y

\{x\} \cdot Y

and

X \cdot \{y\}

are simply represented as

x \cdot Y

and

X \cdot y

, respectively. A hypergroupoid

(N, \cdot)

is called a semihypergroupif for each

x, y, z \in N

(x \cdot y) \cdot z = x \cdot (y \cdot z)

. A semihypergroup

(N, \cdot)

is called a hypergroup if for each

x \in N

x \cdot N = N \cdot x = N

. We denoteidentity elementof a hypergroup N as

0_{N}

for having a addive hyperoperation on N. If the non-additive is in the hypergroup N, we will use the

1_{N}

notation. Let x be any element of

(N, \cdot)

x^{'}

is called an inverse elementof x in N if

0_{N}

\in x \cdot x^{'} \cap x^{'} \cdot x

. The algebraic structure that fulfills the following conditions on H with the hyperoperation + is called acanonical hypergroup:

(i)

(N, +)

is a semihypergroup;

(ii)+ is commutative on

(N, +);

(iii) There is an identity element

0_{N}

(N, +)

;

(iv) There is an inverse element

- x

of x in N such that

0_{N} \in x + (- x)

or simplify

0_{N} \in x - x

(v) If

x \in y + z

, then

y \in x - z

for each

x, y, z \in N

;

Let S be non-empty set having a hyperoperation + and a operation on itself. If

(S, +)

is a canonical hypergroup,

(S, .)

is a semigroup consist of

0_{R}

which is a bilaterally absorbing element, i.e.

0_{S} . a = a . 0_{S} = 0_{S}

for each

a \in S

, and

(b + c) . a = (b . a) + (c . a)

and

a . (b + c) = a . b + a . c

for each

a, b, c \in S

, then S is called a Krasner hyperring. We study on Krasner hyperring with unit element

1_{S}

, where

1_{S} . a = a . 1_{S} = a

for each

a \in S

. Let

(S, +, .)

be a Krasner hyperring and

(N, +)

be a canonical hypergroup with external operation

* : S \times N \to N

. Then N is called a left KrasnerS-hypermoduleif the following conditions hold for each

r_{1}

r_{2} \in S

and for each

x_{1}

x_{2} \in N

(1)

r_{1} * (x_{1} + x_{2}) = r_{1} * x_{1} + r_{1} * x_{2}

;

(2)

(r_{1} + r_{2}) * x_{1} = r_{1} * x_{1} + r_{2} * x_{1}

;

(3)

(r_{1} . r_{2}) * x_{1} = r_{1} * (r_{2} * x_{1})

;

(4)

0_{S} * x_{1} = 0_{N}

A left Krasner S-hypermodule N is called unitary if

1_{S} * x = x

for each

x \in N

. To simplify representation

r x

instead of

r * a

for each

r \in S

and

x \in N

. Throughout the rest of this paper, we assume all hypermodules are left unitary Krasner hypermodules and all hyperrings are Krasner commutative hyperrings. It is a proper generalization of Krasner hypermodules to modules because it carries the rings to Krasner hyperrings.

Let S be a hyperring and J a non-empty subset of S. J is called a left (right) hyperideal if

x - y \subseteq J

and

s x \in J

(x s \in J)

for every

s \in S

x, y \in J

, denoted by

J ⊴_{l} S

(

J ⊴_{s} S

). If J is both left hyperideal and right hyperideal of

S,

denoted by

J ⊴ S

. It is clear that

S . x = \{s . x : s \in S\}

is a hyperideal of S.

Let N be a S-hypermodule and

\emptyset \neq K

\subseteq N

. K is said to be a subhypermodule of N if K is a S-hypermodule itself which is contained in N, denoted by

K \leq N

. Shortly, a non-empty subset K of N is a subhypermodule if

x - y \subseteq K

and

s . x \in K

for every

s \in S

and

x . y \in K

. It is easily demonstrable that

S . x = \{s . x : s \in S\}

is a subhypermodule of a hypermodule N for every

x \in N

. Let K and T be subhypermodules of N. Then

K + T = \{x \in k + t : k \in K, t \in T\}

is a subhypermodule, too. Let N and K be S-hypermodule and

Ψ : N \to K

a function. If

f (x + y) \subseteq f (x) + f (y)

and

f (s . x) = s . f (x)

for every

x, y \in K

and

s \in S

, f is called a hypermodule S-homomorphism from N to K. Instead of this statement if the inclusion satisfies

f (x + y) = f (x) + f (y)

, then f is said to be a strongS-homomorphismfrom N to K. The class of every strong S-homomorphisms from N to K is denoted by

H o m_{S} (N, K)

, sets are defined as

k e r (Ψ) : = \{x \in N : Ψ (x) = 0_{K}\}

I m (Ψ) : = \{y \in K : \exists x \in N, y \in Ψ (x)\}

. The homomorphism

Ψ \in H o m_{S} (N, K)

is called strongly surjective if

f (x_{1}) = f (x_{2})

implies

x_{1} = x_{2}

for every

x_{1}, x_{2} \in N

, and + is called strongly injective if

f (x_{1}) \cap f (x_{2}) \neq \emptyset

implies

x_{1} = x_{2}

for every

x_{1}, x_{2} \in N

. To simplfy denoting annihilator of an S-hypermodule N for a subhypermodule K, we use the symbol

K :_{S} N

, and the set is a hyperideal which is defined as

\{s \in S : s . N \subseteq K\}

. Another representation of

D :_{S} N

A n n_{S} (N)

As a generalization of prime spectrum of the ring of commutative topology defined on S with Zariski topology [14] inspired by the interaction between the theoretical properties of the hyperring S of the text, over a commututive hyperring S on a several hypermodule N, we examine a Zariski topology on these spectrum

χ_{N}

of pseudo-prime subhyper-modules and we give the interaction between topological hyperspace.

We give topological conditions such as connectedness, Noetherianness and irreducibility in the pseudo-prime spectrum of hypermodules and obtain more information about the algebraic hyperstructure of these hypermodules. Further, we prove this topological hyperspace in terms of spectral hyperspace which is a topological hyperspace and is homeomorphic to

S p e c (S)

for any hyperring S.

2. Condition of Pseudo-Prime for Commutative Krasner Hypermodules

In this section, we present pseudo-prime subhypermodules as a new concept of hypermodules theory. Then we investigate connection between spectral hyperspace and Zariski topology. Recall from [5] that a proper hyperideals J of a hyperring S is called prime if

X Y \subseteq J

implies

X \subseteq J

Y \subseteq J

for every hyperideals X, Y of S.

Definition 1.

Let N be an S-hypermodule and K a subhypermodule of N.

(1) We call a subhypermodule K pseudo-prime if

(K :_{S} N)

is a prime hyperideal of S.

(2) We call a pseudo-prime spectrum of N as the set of all pseudo-prime submodules of N, express it by

X_{N}^{S}

or shortly

χ_{N}

. For any prime hyperideal

J \in X_{S} = S p e c (S)

, the collection N of whole pseudo-prime subhypermodules of N with

(K : N) = J

is denoted by

X_{N, J}

(3) For a subhypermodule K of N, we define the set

V^{N} (K) = \{Y \in X_{N} : K \leq Y\}

. If it would be written as shortly, using

V (K)

instead of

V^{N} (K)

(4) If

X_{N} = \emptyset

for every

Y \in X_{N}

, the function

η : X_{N} \to S p e c (S / A n n (N))

defined by

η (Y) = (Y : N) / A n n (N)

is called natural map of

X_{N}

. If

N = (0)

N \neq (0)

, the natural map of

X_{N}

is strongly surjective, then we call N pseudo-primeful.

(5) If the natural map of

X_{N}

is strongly injective, then we call N a pseudo-injective.

According to our above definition prime hyperideals of a hyperring S and the pseudo-prime S-hypermodule of the hypermodule S are the same. It is obtained that the concept of prime hyperideal to hypermodules is a strong notion of the strongly pseudo-prime subhypermodule S. Recall from [25] that a proper subhypermodule K of an S-hypermodule N is called prime if for every hyperideal J of S and every subhypermodule X of N, a hypermodule

[J . K] = \cup \{\overset{p}{\sum_{j = 1}} u_{j} . x_{j} : l \in N, u_{j} \in J and x_{j} \in X, for all j\} \subseteq K

implies

J \subseteq (K : N)

X \subseteq K

So a proper subhypermodule K of an S-hypermodule N is prime if

N / K

is a torsion-free

S / (K : N)

-hypermodule, i.e.

N / K

is a hypermodule on a hyperring S such that the only element destroyed by a non-zero divisor of hyperring

S / (K : N)

is zero. By using Definition 2.1, it can be easily seen that every prime subhypermodule K is apseudo-prime subhypermodule, because

(K : N) \in S p e c (S)

. But in general, the converse assertion is not hold.

Example 1.

Take a

Z

-hypermodule

N = Z \oplus Z \oplus Z

and a subhypermodule

K = 〈(3, 0, 0)〉 = (3, 0, 0) Z

. Then

(K : N) = (0) \in S p e c (Z)

. So K is a prime subhypermodule of N. But

K \in X_{N}

, K isn’t a prime subhypermodule.

Recall from [17] that a hypermodule N is called multiplication S-hypermodule if for each subhypermodule K of N, there is a hyperideal J of S so that

K = [J . N]

Recall from [11] that a proper subhypermodule K of N is called maximal if for each subhypermodule L of N with

K \leq L \leq N

, then

K = L

L = N

Example 2.

Every multiplication hypermodule satisfies the condition pseudo-injective. Take the

Z

-hypermodule

N = Z_{p} \oplus Z (p^{\infty})

where p is a prime integer. Say, K the pseudo-prime subhypermodule of N. It followa that

(K : N) N \subseteq K < N

. Suppose that

(K : N) \neq p

. Then it contadicts with torsion

Z

-hypermodule N. Thus

(K : N)

is equal to p. It is seen clearly that

(K : N) N

is a maximal subhypermodule of N by using the strong isomorphism

Z_{p} ≅ N / P N

. Then we have

(K : N) N = K

. Therefore N is a pseudo-injective

Z

-hypermodule. But there isn’t a hyperideal J of

Z

with

Z_{p} \oplus 〈0〉 = J . N

, N isn’t a multiplication

Z

-hypermodule.

Lemma 1.

The following assertions are equivalent for a finitely generated S-hypermodule

N .

(1) N is a multiplication hypermodule.

(2) N is a pseudo-injective hypermodule.

(3)

|X_{N, J}| \leq 1

for each maximal hyperideal J of S.

(4)

N / J . N

is simple for each maximal hyperideal J of S.

Proof.

(1) \Rightarrow (2)

By Example 2.2

(2) \Rightarrow (3)

Clear by Definition 2.1

(3) \Rightarrow (4)

It can be proven clearly that

J . N = N

for a maximal hyperideal J of S. Hence suppose that

J . N \neq N

and

K / J . N \subset N / J N

. Then K is a proper subhypermodule containing the subhypermodule

J N

of N. Thus we have

J = (J . N : N) = (K : N)

. Since K and

J N

are belong to

X_{N : J}

, then

K = J . N

by the assumption. So

N / J . N

is a simple S-hypermodule. By [17], N is a multiplication hypermodule. □

Further, we use the concept of pseudo-prime subhypermodules to describe another new hypermodule class, namely the topological hypermodule. We give some examples of topological hypermodules and explore some algebraic properties of this hypermodule class. Then in the next section we connect a topology to the set of all pseudo-prime subhypermodules of topological hypermodules, called the Zariski topology. Let L be a subset of

X_{N}

for an S-hypermodule N. We show as notation the intersection of all elements in L by

ℑ (L)

Definition 2.

Let N be an S-hypermodule.

(1) If it is an intersection of pseudo-prime subhypermodules of N, N is said to be pseudo-semiprime.

(2) If

T \cap L \leq K

, then the pseudo-prime subhypermodule K of N is called extraordinary, where

T,

L are pseudo-semiprime subhypermodules of N, then either

L \subseteq K

T \subseteq K

(3) The pseudo-prime radical of K is shown as notation

P r a d (K)

is the intersection of each pseudo-prime subhypermodules of N containing K, i.e.

P r a d (K) = ℑ (V (K)) = \underset{P \in V (K)}{\cap P}

. If

V (K) = \emptyset

, then we get

P r a d (K) = N

for a subhypermodule K of N.

(4)If

K = P r a d (K)

, then the subhypermodule K of N is said to be a pseudo-prime radical subhypermodule.

(5) If

X_{N} = \emptyset

or each pseudo-prime subhypermodule of N is extraordinary then N is said to be topological.

By using Definition 2.2 that we prove that every prime hyperideal of S is extraordinary pseudo-prime subhypermodule for the S-hypermodule S. It is not always true thatevery pseudo-prime subhypermodule is extraordinary. Take a

Z

-hypermodule

Q \oplus Z_{p}

, where

Q

is a rational numbers set as a

Z

-hypermodule and p is a prime integer.

〈0〉 \oplus

Z_{p}

Q \oplus 〈0〉

and

Z \oplus 〈0〉

Q \oplus

Z_{p}

are pseudo-prime subhypermodules. Since

(〈0〉 \oplus Z_{p}) \cap (Q \oplus 〈0〉) \subseteq Z \oplus 〈0〉

Z \oplus 〈0〉

isn’t extraordinary. Hence,

Q \oplus

Z_{p}

isn’t a topological hypermodule. In addition the

Z

-hypermodule

Z (p^{\infty})

is a topological hypermodule as its subhypermodules are linearly ordered where p is a prime integer.

Theorem 1.

Let N be a topological S-hypermodule. Then the following statements hold.

(1)Every strong homomorphic image of N is a topological S-hypermodule.

(2)

N_{J}

is a topological

S_{J}

-hypermodule for every prime ideal J of S.

Proof.

(1) Let K be a subhypermodule of N. We have a factor S-hypermodule

N / K

, say L. Let

U / K

be a pseudo-prime subhypermodule of L. It follows from

(U / K : L) = (U : N)

that U is a pseudo-prime subhypermodule of N. Let

V / K

and

W / K

be pseudo-semiprime subhypermodule of L so that

V / K \cap W / K \subseteq U / K

. So V and W are pseudo-semiprime subhypermodules of N such that

V \cap W \subseteq U

. By the hypothesis,

V \subseteq U

W \subseteq U

. Therefore

V / K \subseteq U / K

W / K \subseteq U / K

. Consequently L is a topological S-hypermodule.

(2) Let L be a pseudo-prime subhypermodule of the

S_{J}

-hypermodule

N_{J}

. Let

Ψ : N \to N_{J}

be the canonical strong homomorphism. Firstly we shall prove that

L \cap N

is a pseudo-prime subhypermodule of N. Let I and

I^{'}

be hyperideals of S so that

I I^{'} \subseteq (L \cap N :_{S} N)

. By using the canonical strong homomorphic image of N by

Ψ

, we have

(I_{J} I_{J}^{'}) . N_{J} \subseteq L = {(L \cap N)}_{J}

. Since L is a pseudo-prime subhypermodule of the

S_{J}

-hypermodule

N_{J}

, either

I_{J} \subseteq (L : N_{J})

I_{J}^{'} \subseteq (L : N_{J})

. So we have

I . N \subseteq {(I . N)}_{J} \cap N \subseteq L \cap K

I : N \subseteq L \cap K

. It follows that

L \cap K

is a pseudo-prime subhypermodule of N. Take a pseudo-semiprime subhypermodules

K_{1}

and

K_{2}

N_{J}

with

K_{1} \cap

K_{2} \subseteq L

. We have

K_{1} \cap

N and

K_{2} \cap

N are pseudo-semiprime subhypermodules of N with

(K_{1} \cap N) \cap (K_{2} \cap N) = (K_{1} \cap K_{2}) \cap N \subseteq L \cap N

that

K_{1} = {(K_{1} \cap N)}_{J} \subseteq {(L \cap N)}_{J} = H

K_{2} = {(K_{2} \cap N)}_{J} \subseteq {(L \cap N)}_{J} = H

. Therefore H is extraordinary and

N_{J}

is a topological

S_{J}

-hypermodule. □

Recall that the pseudo-prime subhypermodules of S as on S-hypermodule are the pseudo-prime hyperideals for any hyperring S. In the following Theorem, we extend the fact into Theorem 2.1 to multiplication hypermodules.

Theorem 2.

Let N be a finitely generated S-hypermodule. Then the following assertions are equivalent.

(1) N is a multiplication hypermodule..

(2) There is a hyperideal J of S so that

V (K) = V (J . K)

for every subhypermodule K of N.

(3) N is a topological hypermodule.

Proof.

(1) ⟹ (2)

Clear

(2) ⟹ (3)

Let L be a pseudo-prime subhypermodule of N, K and U be pseudo-semiprime subhypermodules of N such that

K \cap U \subseteq L

. Then we have

V (K) = V (J . N)

and

V (U) = V (J : N)

for hyperideals J and

J^{'}

of S. Take some collection of pseudo-prime subhypermodules

{\{K_{α}^{'}\}}_{α \in Ω}

such that

K = \underset{α \in Ω}{\cap} K_{α}^{'}

. So we get

(J \cap J^{'}) . N \subseteq K_{α}^{'}

for every

α \in Ω

by using the conclusion

K_{α}^{'} \in V (K) \subseteq V (K) \cup V (U) = V (J . N) \cup V (J^{'} . N = V ((J \cap J^{'}) . N))

. Hence

(J \cap J^{'}) . N \subseteq \underset{α \in Ω}{\cap} K_{α}^{'} = K

. By similar way, we have the conclusion

(J \cap J^{'}) . N \subseteq U

. Thus

(J \cap J^{'}) . N \subseteq K \cap U \subseteq L

. It follows from

J \cap J^{'} \subseteq (L : N)

that

L \in V (J . N) = V (K)

L \in V (J : N) = V (U)

, that is either

K \subseteq L

U \subseteq L

(3) ⟹ (1)

Clear by Lemma 2.4 □

Definition 3.

Let N be an S-hypermodule. Then N is called content if

b \in c (b) N

where

c (b) = \cap \{J : J is a hyperideal of S and b \in J . N\}

for every

b \in N

. It shall be given as an equivalent definition to it. N is a content S-hypermodule if and only if

(\underset{α \in Ω}{\cap} J_{α}) . N = \underset{α \in Ω}{\cap} (J_{α} . N)

for every family of

\{J_{α} : α \in Ω\}

S .

Theorem 3.

Let N be a content and pseudo-injective S-hypermodule. Then N is topological. In addition, if

P r a d (K) = \sqrt{(K : N)} N

for every subhypermodule K of N, N is topological.

Proof.

Let N be a content and pseudo-injective ,

P r a d (L) = N

. Then we have

V (K) = V (S . N)

. If

P r a d (L) \neq N

, then

P r a d (L)

is a pseudo-semiorime subhypermodule of N. So there exist pseudo-prime subhypermodules

L_{α}

for every

α \in Ω

with

P r a d (L) = \underset{α \in Ω}{\cap} L_{α}

and

(L_{α} : N) = p_{α} \in S p e c (S)

. It follow from

p_{α} N = (p_{α} : N) . N = ((p_{α} : N) . N : N)

and N is pseudo-injective for every

α \in Ω

that

L_{α} = p_{α} N

. Since N is a content hypermodule,

P r a d (L) = \underset{α \in Ω}{\cap} L_{α} = \underset{α \in Ω}{\cap} (p_{α} N) = (\underset{α \in Ω}{\cap} p_{α}) N

= \underset{α \in Ω}{\cap} (L_{α} : N) N = (\underset{α \in Ω}{\cap} L_{α} : N) N

= (P r a d (L) : N) N .

Then we obtain

V (L) = V (P r a d (L)) = V ((P r a d (L) : N) N)

. It follows from Theorem 2.2 that N is a topological hypermodule.

Let N be a hypermodule where every subhypermodule K of N satisfies the equality

P r a d (K) = \sqrt{(K : N)} N

. Then

V (K) = V (P r a d (K)) = V (\sqrt{(K : N)} N)

. By using Theorem 2.2, we have N is atopological hypermodule. □

3. PSEUDO-PRIME SPECTRUM OVER TOPOLOGICAL HYPERMODULES

We use denoting N as a topological S-hypermodule in the rest of this text. In [17], we investigate the Zariski topology over multiplication hypermodules. Zariski topology is built on topological modules in [14]. In this section, inspired by this source, this class will be examined in hypermodules by looking at it from a different spectrum. Briefly J and

\bar{J}

will be used instead of

S / A n n (N)

and

J / A n n (N)

for every hiperideal

J \in V^{s} (A n n (N))

Theorem 4.

X_{N}

is connected for a pseudo-primeful S-hypermodule N, then

X_{\bar{S}}

is connected.

Proof.

Let

φ : X_{N} \to S p e c (S / A n n (N))

be a natural map. As

φ

is surjective, we must show that

φ

is continuous. Take a hyperideal J of S containing

A n n (N)

. Let

K \in φ^{- 1} (V^{\bar{S}} (\bar{J}))

. There is a hiperideal

\bar{J^{'}} \in V^{\bar{S}} (\bar{J})

such that

φ (K) = \bar{J^{'}}

. Thus

J \subseteq (K : N) = J

. It follows from

J . N \subseteq K

that

K \in V^{N} (J . N)

. Let

L \in V^{N} (J . N)

. Then we obtain

J \subseteq (J . N : N) \subseteq (L : N)

. Therefore

L \in φ^{- 1} (V^{\bar{S}} (\bar{J}))

φ

is continuous as

φ^{- 1} (V^{\bar{S}} (\bar{J})) = V^{N} (J . N)

. □

In the following proposition, we obtain basic properties of the subhypermodules of N taking the topological hyperspace

X_{N}

is a

T_{1}

-hyperspace.

Proposition 1.

Let

Y \subseteq X_{N}

and

K \in X_{N, J}

for any

J \in S p e c (S)

. Then the following statements hold.

(1)

C l (Y) = V (ℑ (Y))

. Thus

Y = V (ℑ (Y))

⇔Y is closed.

(2)

〈0〉 \in Y

provided that Y is dense in

X_{N}

(3)

X_{N}

is a

T_{0}

-hyperspace.

(4) Every pseudo-prime subhypermodule of N is a maximal element in the set of whole pseudo-prime subhypermodules of N if and only if

X_{N}

is a

T_{1}

-hyperspace.

(5)

S p e c (S)

is a

T_{1}

-hyperspace provided that

X_{N}

is a

T_{1}

-hyperspace.

Proof.

(1) The inclusion

V (ℑ (Y)) \supseteq Y

is clear. Let

V (K)

be any closed subset of

X_{N}

containing Y. Then,

V (ℑ (Y)) \subseteq V (ℑ (V (K))) = V (P r a d (K)) = V (K)

since

ℑ (V (K)) \subseteq ℑ (Y)

. It follows that

V (ℑ (Y))

is the smallest closed subset of

X_{N}

containing Y. Therefore, the equality is obtained.

(2) It can be seen clearly thanks to the condition (1).

(3) To show

X_{N}

is a

T_{0}

-hyperspace, we have to prove that all closures of distinct points in

X_{N}

are distinct. Let H and K be any distinct point of

X_{N}

. By using the condition (1), we have

C l (\{H\}) = V (H) \neq V (K) = C l (\{K\})

, this is also desired.

(4) Topologically, we know that for

X_{N}

to be a

T_{1}

-hyperspace, it must be each singleton subset is closed. Let L be a maximal element in the set of all pseudo-prime subhypermodules of N, by using the condition (1) we get that

C l (\{L\}) = V (L) = \{L\}

. So

\{L\}

is closed. We obtain that

X_{N}

is a

T_{1}

-hyperspace. Conversely, let

\{L\}

be closed as

X_{N}

is a

T_{1}

-hyperspace. Then

\{L\} = C l (\{L\}) = V (ℑ (\{L\})) = V (L)

. So L is a maximal element in the set of whole pseudo-prime subhypermodules of N.

(5) Let L be a pseudo-prime subhypermodule of N. We have

C l (\{L\}) = V (L)

by using the condition (1). Let

H \in V (L)

. By the hypothesis, we have

(L : N) = (H : N) \in M a x (S)

. Thus, L and H are prime subhypermodule of N. By Theorem 2.2,

H = L

. It follows from

C l (\{L\}) = L

that

X_{N}

is a

T_{1}

-hyperspace. □

Definition 4.

A topological hyperspace N is called irreducible if for every decomposition

N = N_{1} \cup N_{2}

as closed subsets

N_{1}

and

N_{2}

of N provided that

N_{1} = N

N_{2} = N

. In addition, a maximal irreducible subset of N is said to be an irreducible component of the topological hyperspace N.

The next theorem reveals the relation between pseudo-prime subhypermodules of the S-hypermodule N and irreducible subset of the topological hyperspace

X_{N}

. It is clear that for a hyperring S, a subset K of

S p e c (S)

is irreducible ⇔

ℑ (K)

is a prime hyperideal of S.

Theorem 5.

Let N be an S-hypermodule and K be a subset of

X_{N}

. Then

ℑ (K)

is a pseudo-prime subhypermodule of N⇔K is an irreducible hyperspace.

Proof.

(\Rightarrow)

Let K be an irreducible hyperspace, T and U be hyperideals of S with

T U \subseteq (ℑ (K) : N)

. Then we have

K \subseteq V (ℑ (K)) \subseteq V ((T U) . N) = V (T . N) = \cup V (U . N)

. It follows from K is irreducible that we have

K \subseteq V (T . N)

K \subseteq V (U . N)

. So

T . N \subseteq P r a d (T . N) = ℑ (V (U . N)) \subseteq ℑ (N)

U . N \subseteq ℑ (K)

. Since

T \subseteq (ℑ (K) : N)

U \subseteq (ℑ (K) : N)

, then

ℑ (K)

is a pseudo-prime subhypermodule of N. Let’s take a pseudo-prime subhypermodule

ℑ (K)

of N with

K \subseteq K_{1} \cup K_{2}

where

K_{1}

and

K_{2}

are closed subsets of

X_{N}

. Thus there exist subhypermodules L and T of N such that

V (L) = K_{1}

and

V (T) = K_{2}

. Therefore

ℑ (V (L) \cup V (T)) = ℑ (V (L)) \cap ℑ (V (T)) = P r a d (L) \cap P r a d (T) \subseteq ℑ (K)

. Then we have

ℑ (K)

is an extraordinary subhypermodule because N is a topological hypermodule. It is obtained that

P r a d (L) \subseteq ℑ (L)

P r a d (T) \subseteq ℑ (K)

. Then K is irreducible provided that

K \subseteq V (ℑ (K)) \subseteq V (P r a d (L)) = V (L) = K_{1}

K \subseteq K_{2}

. □

Corollary 1.

Let N be an S-hypermodule and K be a subhypermodule of N.

(1)

P r a d (K)

is a pseudo-prime subhypermodule of

N \Leftrightarrow V (K)

is an irreducible hyperspace.

(2)

P r a d (0)

is a pseudo-prime subhypermodule of N if and only if N is a irreducible hyperspace.

(3) If

X_{N, U} \neq \emptyset

for any

u \in S p e c (S)

, then

X_{N, U}

is an irreducible hyperspace.

Proof.

(1) It follows from

P r a d (K) = ℑ (V (K))

that the proof is obtained directly using Theorem 3.2.

(2) Clear from (1) by taking

K = (0)

(3) Since

ℑ (X_{N, U} : N) = \underset{Q \in X_{N, U}}{\cap} (Q : N) = U \in S p e c (S)

, the claim provides thanks to Theorem 3.2. □

Definition 5.

Let N be an S-hypermodule, U a hyperideal of N. U is said to be a radical hyperideal of S if

U = \underset{i}{\cap} u_{i}

where

u_{i}

runs through

V^{S} (U)

Lemma 2.

Let N be a non-zero pseudo-primeful S-hypermodule, U a radical hyperideal of S. Then

A n n (N) \subseteq U

if and only if

(U . N : N) = U

. In addition, for every

u_{i} \in V^{S} (A n n (N))

u_{i} . N

is a pseudo-prime subhypermodule of N.

Proof.

(\Rightarrow)

Clear.

(\Leftarrow)

By the hypothesis, there exists hyperideals U where

u_{i}

runs through

V^{S} (U)

and

A n n (N) \subseteq U = \underset{i}{\cap} u_{i}

. Then there is a pseudo-prime subhypermodule

K_{i}

of N with

(K_{i} : N) = u_{i}

for a pseudo-primeful S-hypermodule N and

u_{i} \in V^{S} (A n n (N))

. So we have

U \subseteq (U . N : N) = ((\underset{i}{\cap} u_{i}) . N : N) \subseteq \underset{i}{\cap} (u_{i} . N : N) \subseteq \underset{i}{\cap} (K_{i} : N) = \underset{i}{\cap} u_{i} = U

. Hence

(U . N : N) = U

. □

Recall from [13] that

R a d (N)

is the sum of all small subhypermodules of N, that is

R a d (N) = \sum_{L_{i} < < N} L_{i}

. Here subhypermodules

L_{i}

of N is called small in N if

T + T_{i} = N

for every subhypermodule T of N satisfies

N = T

Now let’s adapt the Nakayama’s Lemma to hypermodule in the next proposition.

Proposition 2.

Let N be a pseudo-primeful S-hypermodule and U a hyperideal of S which contained in

R a d (S)

so that

U . N = N

. Then

N = (0)

Definition 6.

Let T be closed subset of a topological hyperspace. If

T = C l (\{a\})

then

a \in Y

is said to be the generic point of T.

In Proposition 3.1 (1) we obtain that each element K of

X_{N}

is a general point of the irreducible closed subset

V (K)

. Note that if the topological hyperspace

T_{0}

-hyperspace, the general point T of a closed subset of the topological hyperspace is unique by Proposition 3.1. The following theorem is an excellent implementation of Zariski topology on hypermodules. Indeed, the following theorem shows that there is a relationship between the irreducible closed subsets of

X_{N}

and the pseudo-prime subhypermodules of the S-hypermodule N.

Theorem 6.

Let N be a S-hypermodule and

U \subseteq X_{N}

. Then the following conditions satisfy.

(1) U is an irreducibleclosed subset of

X_{N}

if and only if

U = V (W)

for each

W \in X_{N}

. In addition each irreducible closet subset of

X_{N}

possesses a generic point.

Recall from [5] that a hyperring S is said to be Noetherian if it satisfies the ascending chain condition on hyperideals of S, i.e., for each ascending chain of hyperideals

J_{1} \subseteq J_{2} \subseteq . . .,

there is an element

k \in N

such that

J_{k} = J_{t}

for every

k \geq t

Definition 7.

A topological hyperspace X is said to be Noetherian hyperspace if the open subset of the hyperspace possesses the ascending chain condition.

We use the notion of Noetherian S-hypermodules for pseudo-prime spectrum of hypermodules and radical hyperideals of S satisfing the ascending chain condition ACC.

Theorem 7.

Let N be a S-hypermodule. Then N possesses Noetherian pseudo-prime spectrum ⇔ the ACC is provided pseudo-prime radical subhypermodules of N.

Proof.

(\Rightarrow)

Let N has a Noetherian pseudo-prime spectrum and

U_{1} \subseteq U_{2} \subseteq . . .

an ascending chain of pseudo-prime radical subhypermodules of N. Hence

U_{j} = ℑ ((V (U_{j}))) = P r a d (U_{j})

for

j \in N

. It follows that

V (U_{1}) \supseteq V (U_{2}) \supseteq . . .

is a descending chain of close subset of

X_{N}

. By the hypothesis there exists an element

l \in N

so that

V (U_{l}) = V (U_{l + n})

for each

n \in N

. So

N_{1} = P r a d (U_{l}) = ℑ (V (U_{l})) = ℑ (V (U_{l + n})) = P r a d (U_{l + n}) = U_{l + n}

(\Leftarrow)

Suppose that the ACC is provided for pseudo-prime radical subhypermodules of N. Let

V (U_{1}) \supseteq V (U_{2}) \supseteq . . .

be a descending chain of close subsets of

X_{N}

for

U_{j} \leq N

. Then

ℑ (V (U_{1})) \subseteq ℑ (V (U_{2})) \subseteq . . .

is an ascending chain of psudo-prime radical subhypermodules

ℑ (V (U_{j})) = P r a d (U_{j})

of the hypermodule N. By the hypothesis, there is an element

l \in N

so that

ℑ (V (U_{l})) = ℑ (V (U_{l + j}))

for each

j \in N

. It follows from Proposition 3.1 that

V (U_{l}) = V (ℑ (V (U_{l}))) = V (ℑ (V (U_{l + j}))) = V (U_{l + j})

. So

X_{N}

is a Noetherian hyperspace. □

Definition 8.

A topological hyperspace Y is a spectral hyperspace if it is homeomorphic to

S p e c (S)

where S is a hyperring according to the Zariski topology.

Theorem 8.

Let N be a S-hypermodule. Then

X_{N}

is a spectral hyperspace if each of the following conditions are met.

(1) There exists a hyperideal J of S so that

V (U) = V (J . N)

for a Noetherian hyperring S and for every subhypermodule U of N.

(2) Let N be a content pseudo-injective S-hypermodule and

S p e c (S)

a Noetherian topological hyperspace.

Proof.

(1) If it is shown that every subset of

X_{N}

is quasi-compact, the desired is obtained. Let K be an open subset of

X_{N}

and

{\{A_{i}\}}_{i \in N}

be an open coverof K. Then there exist subhypermodules L and

L_{i}

so that

K = X_{N} ∖ V (L)

A_{i} = X_{N} ∖ V (L_{i})

for every

i \in I

and

K \subseteq \underset{i \in I}{\cup} A_{i} = X_{N} ∖ \underset{i \in I}{\cap} V (L_{i})

. By assumption, there is a hyperideal

I_{i}

in S so that

V (L_{i}) = V (I_{i} . N)

for every

i \in I

. Then we have

L \subseteq X_{N} ∖ V (\sum_{i \in I} I_{i} . N) = X_{N} ∖ V ((\sum_{i \in I} I_{i}) . N)

. As S is a Noetherian hyperring, there is a finite subset

I^{'}

of I so that

L \subseteq \underset{j \in I^{'}}{\cup} A_{j}

. Hence

X_{N}

is a both of Noetherian hyperspace and spectral hyperspace.

(2)Let’s show that

X_{N}

is Noetherian. Let

V (L_{1}) \supseteq V (L_{2}) \supseteq . . .

be a descending chain of closed subsets of

X_{N}

. So

P r a d (L_{1}) \subseteq P r a d (L_{2}) \subseteq . . .

S p e c (S)

is Noetherian, the ACC

(P r a d (L_{1}) : N) \subseteq (P r a d (L_{2}) : N) \subseteq . . .

of radial hyperideals shall be stationary by Theorem 3.5. Therefore there exists an element

l \in N

so that

(P r a d (L_{l}) : N) = (P r a d (L_{l + j}) : N) = . . .

, for every

j = 1, 2, . . .

If the proof technique in Theorem 2.3 is applied, it is seen that

P r a d (L_{j}) = (P r a d (L_{j}) : N) . N

. Thus, we get

P r a d (L_{l}) = P r a d (L_{l + j}) = . . .

for every

j = 1, 2, . . .

It follows that

V (L_{l}) = V (P r a d (L_{l})) = V (P r a d (L_{l + j})) = V (L_{l + j}) = . . .

X_{N}

is Noetherian, the desired is achieved. □

Informed Consent Statement

Not applicable.

Acknowledgments

The authors would like to thank the referees who contributed to the development of the article.

Conflicts of Interest

The authors declare that they have no competing interests.

References

R. Ameri, M. Aivazi, S. Hoskova-Mayerova, Superring of polynomials over a hyperring, Mathematics, 7(902), (2019) 1-15. [CrossRef]
R. Ameri, A categorical connection between categories (m, n)-hyperrings and (m, n)-rings via the fundamental relation, Kraguje- 365 vac Journal of Mathematics 45(3), (2019) 361-377. [CrossRef]
R. Ameri, On categories of hypergroups and hypermodules, J. Discrete Math. Sci. Cryptography, 6(2-3), (2003) 121-132. [CrossRef]
D. D. Anderson, Y. Al-shaniafi, Multiplication modules and the ideal, Comm. Algebra, 30(7), 3383 – 3390 (2002). [CrossRef]
H. Bordbar, I. Cristea, Height of prime hyperideals in Krasner hyperrings, Filomat, 31(19), 6153 – 6163 (2017). [CrossRef]
H. Bordbar, M. Novak, I. Cristea, A note on the support of a hypermodule, Journal of Algebra and Its Applications, (2019). [CrossRef]
P. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani editore, Tricesimo, 1993.
P. Corsini, V. Leoreanu-Fotea, Applications of Hyperstructures Theory, Kluwer Academic Publisher, Dordrecht, 2003. [CrossRef]
B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., 2013. [CrossRef]
B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, 2007.
F. Farzalipour and P. Ghiasvand, On secondary subhypermodules, Journal of Algebra and Related Topics, 9(1), (2021), 143-158. [CrossRef]
A. R. M. Hamzekolaee, M. Norouzi, A hyperstructural approach to essentially, Comm. Algebra, 46(11), 4954 –4964 (2018). [CrossRef]
A. R. M. Hamzekolaee, M. Norouzi, V. Leoreanu-Fotea, A new approach to smallness in hypermodules, Algebraic Structures and their Applications, 8(1), (2021), 131-145. [CrossRef]
D. Hassanzadeh-Lelekaami, H. Roshan-Shekalgaurabi, Pseudo-prime submodules of modules, Math. Reports 18(68), (2016), 591-608.
N. Jafarzadeh, R. Ameri, On exact category of (m, n)-ary hypermodules, Cregories and General Algebraic Structures with 382 Applications, 12(1), (2020) 69-88.
M. Krasner, A class of hyperrings and hyperfields, Internat. J. Math. and Math. Sci., 6, (1983), 307-311. [CrossRef]
Ö. Kulak, B. Nişancı Türkmen, Zariski topology over multiplication krasner hypermodules, 74(4), Ukrains’kyi Matematychnyi Zhurnal, 2022, 525-533.
D. Lu and W. Yu, On prime spectrum of commutative rings. Comm. Algebra 34 (2006), 2667–2672. [CrossRef]
F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandenaves, Stockholm, (1934) 45-49.
Ch. G. Massouros, Constructions of hyperfields, Matematica Balkanica, 5, (1991) 250-257.
Ch. G. Massoures, Methods of constructing hyperfields, Internat. J. Math. Sci., 8(4), (1985) 725-728. [CrossRef]
Ch. G. Massoures, Free and cyclic hypermodules, Annali di Matematica Pura ed Applicata, 4, (1988) 153-166. [CrossRef]
R. L. McCasland, M. E. Moor, On the radicals of submodules of finitely generated modules, Canad. Math. Bull., 29, 37 – 39 (1986). [CrossRef]
N. Schwartz and M. Tressl, Elementary properties of minimal and maximal points in Zariski spectra. J. Algebra 323 (2010), 2667–2672. [CrossRef]
A. Siraworakun, S. Pianskool, Characterizations of prime and weakly prime subhypermodules, International Mathematical Forum, 7(58), 2012, 2853-2870.
T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, Inc., Palm Harber, USA, 1994.
G. Zhang, W. Tong, and F. Wang, Spectrum of a noncommutative ring. Comm. Algebra 34 (2006), 8, 2795–2810. [CrossRef]

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The Spectrum of the Zariski Topology for Multiplication Krasner Hypermodules

Abstract

1. Introduction

2. Condition of Pseudo-Prime for Commutative Krasner Hypermodules

3. PSEUDO-PRIME SPECTRUM OVER TOPOLOGICAL HYPERMODULES

Informed Consent Statement

Acknowledgments

Conflicts of Interest

References

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