By

Ameer Al-Abayechi

PHD Student, University of Debrecen

Ameerm.hasan@uokufa.edu.iq 1

This presentation has two Subject

* Near Prime Spectrum

* Pre-Open Sets In Minimal Bitopological Spaces

2

* Near Prime Spectrum

In algebraic geometry and commutative algebra, the Zariski

topology is a topology on algebraic varieties, introduced primarily by

Oscar Zariski a Russian-born American mathematician . And later

generalized for making the set of prime ideals of a commutative ring a

topological space, called the spectrum of the ring. 3

The generalization of the Zariski topology introduced by Hilbert

who suggested defining the Zariski topology on the set of the maximal

ideals of a commutative ring as the topology such that a set of

maximal ideals is closed if and only if it is the set of all maximal ideals

that contain a given ideal. Thus the Zariski topology on the set of

prime ideals (spectrum) of a commutative ring is the topology such

that a set of prime ideals is closed if and only if it is the set of all

prime ideals that contain a fixed ideal. 4

Introduction

Let 𝑹 be a ring with identity . The theory of the prime

spectrum of 𝑹 where 𝑺𝒑𝒆𝒄(𝑹) = *𝑷 ∶ 𝑷 𝒊𝒔 𝒂 𝒑𝒓𝒊𝒎𝒆 𝒊𝒅𝒆𝒂𝒍 𝒐𝒇 𝑹+ has

been developed since 1930 . The modern theory was developed by

Jacobson and Zariski mainly . The topology was defined on 𝑺𝒑𝒆𝒄(𝑹) is

the collection of closed sets to be 𝑽 𝑰 = 𝑷 ∈ 𝑺𝒑𝒆𝒄 𝑹 : 𝑰 ⊆ 𝑷 it is

called the Zariski topology on 𝑺𝒑𝒆𝒄(𝑹) . 5

6

In this paper , we expanded Zariski

topology by introducing a new generalization of

near-ring using the near completely prime ideal

, called near Zariski topology .

7

Definition :-

Let 𝑁 be a nonempty set with two binary operations

(+) , (. ) . (𝑁, +, . ) is called near-ring if and only if :

1. (𝑁, +) is a group (not necessarily commutative) .

2. (𝑁, . ) is a semi group .

3. For 𝑎𝑙𝑙 𝑛1, 𝑛2, 𝑛3 ∈ 𝑁 ; 𝑛1 + 𝑛2 . 𝑛3 = 𝑛1. 𝑛3

+ 𝑛2. 𝑛3(right distributive law) . This near-ring will be

termed as right near-ring .

8

Remarks :-

• If 𝑛1. 𝑛2 + 𝑛3 = 𝑛1. 𝑛2 + 𝑛1. 𝑛3instead of condition (3)

the set N satisfies , then we call 𝑁 a left near-ring .

• If 1. 𝑛 = 𝑛 (𝑛. 1 = 𝑛) then 𝑁 has a left identity(right

identity ) .

• If (𝑁, +) is abelian , we call 𝑁 an abelian near-ring .

• If (𝑁, . ) is commutative we call 𝑁 itself a commutative

near-ring . Clearly if 𝑁 is commutative near-ring then left

and right distributive law is satisfied and 1. 𝑛 = 𝑛. 1 = 𝑛 ,

𝑁 is called unital commutative near-ring .

9

Example :-

Near-ring arise very naturally in the study of

mappings on groups . If (𝐺, +) is a group (not

necessarily abelian ) then the set 𝑀(𝐺) of all

mappings from 𝐺 to 𝐺 is a near-ring with respect to

pointwies addition and composition of mappings .

10

Definition :- Let 𝑁 be a near-ring and 𝐼 is a nonempty subset of 𝑁 . (𝐼, +, . ) is called an ideal of 𝑁 if and only if : 1. (𝐼, +) is normal subgroup of (𝑁, +) . 2. 𝐼𝑁 ⊆ 𝐼𝑁 and for all 𝑛, 𝑛1 ∈ 𝑁 and for all

𝑖 ∈ 𝐼 , 𝑛 𝑛1 + 𝑖 − 𝑛𝑛1 ∈ 𝐼 . Definition :- An ideal 𝑃 of 𝑁 is said to be completely prime if 𝑎𝑏 ∈ 𝑃 implies 𝑎 ∈ 𝑃 or 𝑏 ∈ 𝑃 for any 𝑎, 𝑏 ∈ 𝑁 .

Let 𝑁 be a commutative near-ring with identity and

𝑃 be a completely prime ideal of 𝑁 . The set of all

completely prime ideal of 𝑁 , is denoted by 𝑆𝑝𝑒𝑐(𝑁) is

called near prime spectrum on completely prime ideal . Let 𝐼

is ideal of 𝑁 and let 𝑉(𝐼) collection of all completely prime

ideal contains 𝐼 . The collection of all 𝑉(𝐼) satisfies the

axioms of closed sub sets of a topology for 𝑆𝑝𝑒𝑐(𝑁) called

the near Zariski topology for 𝑆𝑝𝑒𝑐(𝑁) . 11

Theorem 1 :- The near prime spectrum 𝑆𝑝𝑒𝑐(𝑁) of any commutative near-ring 𝑁 is a compact topological space . Corollary 2 :- Let 𝐼 be a near ideal of near-ring 𝑁 . Then the closed subset 𝑉(𝐼) of 𝑆𝑝𝑒𝑐(𝑁) is a compact set .

12

Theorem 3 :-

Let 𝜑: 𝑁1 → 𝑁2 be a near-ring unital homomorphism between

near-rings 𝑁1 and 𝑁2 . Then 𝜑:𝑁1 → 𝑁2 induces a continuous map

𝜑∗: 𝑆𝑝𝑒𝑐(𝑁2) → 𝑆𝑝𝑒𝑐(𝑁1) , where 𝜑∗ 𝑃 = 𝜑−1(𝑃) for every

completely prime ideal 𝑃 of 𝑁2 .

Proof :- Let 𝑃2 be a completely prime ideal of 𝑁2. Now 12 ∉ 𝑃2,

because 𝑃2 is a proper ideal of 𝑁2, then 11 ∉ 𝜑−1 𝑃2 , since

𝜑 11 = 12. It follows that 𝜑−1(𝑃2) is a proper ideal of 𝑁1 . Let 𝑥 and

𝑦 be elements of 𝑁1. Suppose that 𝑥𝑦 ∈ 𝜑−1 𝑃2 . Then 𝜑(𝑥)𝜑(𝑦)

= 𝜑(𝑥𝑦) and therefore 𝜑 𝑥 𝜑 𝑦 ∈ 𝑃2. But 𝑃2 is a completely prime

ideal of 𝑁2, and therefore either 𝜑 𝑥 ∈ 𝑃2 or 𝜑 𝑦 ∈ 𝑃2 .Thus either

𝑥 ∈ 𝜑−1 𝑃2 or 𝑦 ∈ 𝜑−1 𝑃2 . This shows that 𝜑−1 𝑃2 is a completely

prime ideal of 𝑁1.

13

We conclude that there is a well-defined function

𝜑∗: 𝑆𝑝𝑒𝑐 𝑁2 → 𝑆𝑝𝑒𝑐 𝑁1 such that 𝜑∗ 𝑃2 = 𝜑−1 𝑃2 for all

completely prime ideal 𝑃2 of 𝑁2. Now we prove that 𝜑∗ is a

continuous function , let 𝐼1 be a ideal of 𝑁1 ,

𝜑∗−1 𝑉 𝐼1 = 𝑃2 ∈ 𝑆𝑝𝑒𝑐 𝑁2 : 𝜑∗ 𝑃2 ∈ 𝑉 𝐼1 .

since 𝜑∗ 𝑃2 = 𝜑−1 𝑃2

= 𝑃2 ∈ 𝑆𝑝𝑒𝑐 𝑁2 : 𝜑−1 𝑃2 ∈ 𝑉 𝐼1

= 𝑃2 ∈ 𝑆𝑝𝑒𝑐 𝑁2 : 𝐼1 ⊂ 𝜑−1 𝑃2

= 𝑃2 ∈ 𝑆𝑝𝑒𝑐 𝑁2 : 𝜑 𝐼1 ⊂ 𝑃2 = 𝑉 𝜑 𝐼1

Thus , 𝜑∗: 𝑆𝑝𝑒𝑐 𝑁2 → 𝑆𝑝𝑒𝑐 𝑁1 is a continuous function .

14

Theorem 4 :-

Let 𝑁 be a near-ring , let 𝐼 be a proper ideal of 𝑁 ,

and let 𝜋𝐼 : 𝑁 → 𝑁/𝐼 be the corresponding quotient near-

ring homomorphism onto the quotient near-ring 𝑁/𝐼 . Then

the induced map 𝜋𝐼∗: 𝑆𝑝𝑒𝑐 𝑁 𝐼 → 𝑆𝑝𝑒𝑐 (𝑁) maps

𝑆𝑝𝑒𝑐 𝑁 𝐼 homeomorphically onto the closed set 𝑉(𝐼) .

Lemma 5 :-

If 𝑁 is a noetherian near-ring . Then 𝑆𝑝𝑒𝑐(𝑁) is a

noetherian space .

15

Lemma 6 :- Let 𝑁 be a near-ring . Then the space 𝑆𝑝𝑒𝑐(𝑁) is a 𝑇0 space . Proof :- Let 𝑃1 , 𝑃2 ∈ 𝑆𝑝𝑒𝑐 𝑁 and 𝑃1 ≠ 𝑃2 then 𝑃1 ⊈ 𝑃2 or 𝑃2 ⊈ 𝑃1 , let 𝐻 𝑓 = 𝑃1 ∈ 𝑆𝑝𝑒𝑐 𝑁 : 𝑓 ∉ 𝑃1 and 𝑃1 ⊈ 𝑃2. Then We get 𝑃1 ∈ 𝐻 𝑓 , 𝑃2 ∉ 𝐻 𝑓 . Thus 𝑆𝑝𝑒𝑐 𝑁 is a 𝑇0 space . Lemma 7 :- Let 𝑁 be a near-ring . Then the space 𝑆𝑝𝑒𝑐(𝑁) is 𝑇1 if and only if 𝑆𝑝𝑒𝑐(𝑁) = 𝑀𝑎𝑥(𝑁) is the set of all near maximal ideal of 𝑁 . 16

* PRE-OPEN SETS IN MINIMAL BITOPOLOGICAL SPACES

17

Bitopological space started by Kelly 1968 is defined as

follows , a bitopological space is a set endowed with two

topologies. Typically, if the set is 𝑋 and the topologies

are 𝜎 and 𝜏 then the bitopological space is referred to as

(𝑋 , 𝜎, 𝜏)

The notion of pre-open set was introduced by M. E. Abd

El-Monsef et al. 1982 .

The concept of minimal structure was introduced by V.

Popa and T. Noiri in 2000 18

19

In this paper, we expanded bitopological by

introducing a new generalization, which uses a

minimal structure, called minimal bitopological

space . Then we studied pre-open sets in

minimal bitopological spaces with some results

and definitions of separation axioms on the

minimal bitopological .

Let 𝑋 be a nonempty set and let 𝑀 ⊆ 𝑃 𝑋 we say that 𝑀

is minimal structure on 𝑋 if ∅, 𝑋 ∈ 𝑀.

Let 𝑋 be a non-empty set, 𝜏 be a topology on 𝑋, let 𝑀 be a

minimal structure on 𝑋 then the triple (𝑋, 𝜏,𝑀) is called a

minimal bitopological space.

20

21

Let (𝑋, 𝜏, 𝑀) be a minimal bitopological space and 𝐴 be a

sub set of 𝑋, 𝐴 is called pre-open with respect to 𝜏 and 𝑀

, if 𝐴 ⊆ 𝑖𝑛𝑡𝜏(𝑐𝑙𝑀(𝐴)), where 𝑐𝑙𝑀(𝐴) is the closure of 𝐴

with respect to 𝑀 .

Let (𝑋, 𝜏, 𝑀) be a minimal bitopological space , and 𝐴 ⊆ 𝑋

, the intersection of all pre-closed sets containing 𝐴 is

called pre-closure of 𝐴, and is denoted by 𝑃𝑟𝑀𝜏 − 𝐶𝑙 A .

That is 𝑃𝑟𝑀𝜏 − 𝐶𝑙 A is the smallest pre-closed set in 𝑋

containing 𝐴, also 𝐴 is pre-closed if and only if 𝑃𝑟𝑀𝜏

− 𝐶𝑙 A = 𝐴 .

22

Theorem 1 :-

Let (𝑋, 𝜏, 𝑀) be a minimal bitopological space, 𝑌 ⊆ 𝑋,

𝑌 open with respect to 𝜏, 𝑀 and let (𝑌, 𝜏𝑌 , 𝑀𝑌) be a sub

space of (𝑋, 𝜏, 𝑀). If 𝐺 be pre-open in 𝑋 with respect to 𝜏 ,

𝑀 . Then 𝐺 ∩ 𝑌 is pre-open in 𝑌 with respect to 𝜏𝑌 , 𝑀𝑌 .

Corollary 2 :-

Let (𝑋,𝜏,𝑀) be a minimal bitopological space, let

(𝑌, 𝜏𝑌 , 𝑀𝑌) be a sub space of (𝑋, 𝜏, 𝑀) and let 𝑌open with

respect to 𝜏 , 𝑀 . If 𝐺 be pre-closed in 𝑋 with respect to 𝜏 ,

𝑀 . Then 𝐺∩𝑌 is pre-closed in 𝑌 with respect to 𝜏𝑌 , 𝑀𝑌 . 23

Theorem 3 :-

A minimal bitopological space (𝑋, 𝜏, 𝑀) is 𝑃𝑟 − 𝑇𝑜-space if

and only if 𝑃𝑟𝑀𝜏 -closure of distinct points are distinct .

Proof :- Let 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦 implies 𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑥 ≠ 𝑃𝑟𝑀

𝜏 − 𝐶𝑙(*𝑦+).

Since 𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑥 ≠ 𝑃𝑟𝑀

𝜏 − 𝐶𝑙(*𝑦+) there exists at least one point 𝑧,

such that 𝑧 ∈ 𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑥 , but 𝑧 ∉ 𝑃𝑟𝑀

𝜏 − 𝐶𝑙 𝑦 . We claim that

𝑥 ∉ 𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑦 . For, let 𝑥 ∈ 𝑃𝑟𝑀

𝜏 − 𝐶𝑙 𝑦 . Then 𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑥

⊂ 𝑃𝑟𝑀𝜏 − 𝐶𝑙(*𝑦+) which is a contradiction that 𝑥 ∉ 𝑃𝑟𝑀

𝜏 − 𝐶𝑙 𝑦 .

Hence 𝑥 ∈ 𝑋\𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑦 but 𝑃𝑟𝑀

𝜏 − 𝐶𝑙 𝑦 is 𝑃𝑟𝑒-closed, so

𝑋\𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑦 is 𝑃𝑟𝑒-open which contains 𝑥 but not 𝑦. It follows

that (𝑋, 𝜏, 𝑀) is 𝑃𝑟 − 𝑇𝑜-space.

24

25

Conversely,

since (𝑋, 𝜏, 𝑀) is 𝑃𝑟 − 𝑇𝑜-space, then for each two

distinct points 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦 there exists 𝑃𝑟𝑒-open set 𝐺

such that 𝑥 ∈ 𝐺, 𝑦 ∉ 𝐺 . 𝑋\𝐺 is 𝑃𝑟𝑒-closed set which does

not contain 𝑥 but contains 𝑦. By Definition , 𝑃𝑟𝑀𝜏 − 𝐶𝑙(*𝑦+)

is the intersection of all 𝑃𝑟𝑒-closed sets which contain *𝑦+.

Thus , 𝑃𝑟𝑀𝜏 − 𝐶𝑙(*𝑦+) ⊂ 𝑋\𝐺, then 𝑥 ∉ 𝑋\𝐺. This implies

that 𝑥 ∉ 𝑃𝑟𝑀𝜏 − 𝐶𝑙(*𝑦+) , but 𝑥 ∈ 𝑃𝑟𝑀

𝜏 − 𝐶𝑙(*𝑥+) , 𝑥 ∉ 𝑃𝑟𝑀𝜏

− 𝐶𝑙(*𝑦+). Therefore 𝑃𝑟𝑀𝜏 − 𝐶𝑙 𝑥 ≠ 𝑃𝑟𝑀

𝜏 − 𝐶𝑙 𝑦 .

Theorem 4 :- Every subspace of 𝑃𝑟 − 𝑇𝑜-

space is 𝑃𝑟 − 𝑇𝑜-space .

Theorem 5 :- Every subspace of 𝑃𝑟 − 𝑇1-

space is 𝑃𝑟 − 𝑇1-space .

Theorem 6 :- Every subspace of 𝑃𝑟 − 𝑇2-

space is 𝑃𝑟 − 𝑇2-space .

Theorem 7 :- Every subspace of a 𝑃𝑟𝑒-

regular space is a 𝑃𝑟𝑒-regular space.

Theorem 8 :- Every 𝑃𝑟𝑒-closed subspace

of a 𝑃𝑟𝑒-normal space is a 𝑃𝑟𝑒-normal.

26

Theorem 9 :-

A minimal bitopological space (𝑋, 𝜏, 𝑀) is an 𝑃𝑟 − 𝑇1-

space If and only if every singleton subset *𝑥+ of is 𝑃𝑟𝑒-

closed set.

Proof :- Suppose 𝑋 is an 𝑃𝑟 − 𝑇1-space. Then for 𝑥, 𝑦 ∈ 𝑋

and 𝑥 ≠ 𝑦, there are two 𝑃𝑟𝑒-open sets 𝐺 and 𝐻 such that

𝑥 ∈ 𝐻, 𝑦 ∉ 𝐻 and 𝑦 ∈ 𝐺 and 𝑥 ∉ 𝐺. So 𝐺 ⊂ *𝑥+𝑐 also

∪ *𝐺 ∶ 𝑦 ≠ 𝑥+ ⊆ *𝑥+𝑐 and *𝑥+𝑐⊆∪ *𝐺: 𝑦 ≠ 𝑥+ . Hence

*𝑥+𝑐=∪ *𝐺: 𝑦 ≠ 𝑥+, which is a 𝑃𝑟𝑒-open set. Then *𝑥+ is a

𝑃𝑟𝑒-closed . Conversely, Let 𝑥, 𝑦 ∈ 𝑋 and 𝑥 ≠ 𝑦. If *𝑥+ and

*𝑦+ are the 𝑃𝑟𝑒-closed sets of 𝑥 and 𝑦 respectively such that

𝑥 ≠ *𝑦+, then *𝑥+𝑐 and *𝑦+𝑐 are 𝑃𝑟𝑒-open sets such that

𝑦 ∈ *𝑥+𝑐and 𝑥 ∉ *𝑥+𝑐also 𝑥 ∈ *𝑦+𝑐and 𝑦 ∉ *𝑦+𝑐. Then 𝑋 is

an 𝑃𝑟 − 𝑇1-space . 27

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