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 ๐น be a ring with identity . The theory of the prime
spectrum of ๐น where ๐บ๐๐๐(๐น) = *๐ท โถ ๐ท ๐๐ ๐ ๐๐๐๐๐ ๐๐ ๐๐๐ ๐๐ ๐น+ has
been developed since 1930 . The modern theory was developed by
Jacobson and Zariski mainly . The topology was defined on ๐บ๐๐๐(๐น) is
the collection of closed sets to be ๐ฝ ๐ฐ = ๐ท โ ๐บ๐๐๐ ๐น : ๐ฐ โ ๐ท it is
called the Zariski topology on ๐บ๐๐๐(๐น) . 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 .
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Definition :-
Let ๐ be a nonempty set with two binary operations
(+) , (. ) . (๐, +, . ) is called near-ring if and only if :
1. (๐, +) is a group (not necessarily commutative) .
2. (๐, . ) is a semi group .
3. For ๐๐๐ ๐1, ๐2, ๐3 โ ๐ ; ๐1 + ๐2 . ๐3 = ๐1. ๐3
+ ๐2. ๐3(right distributive law) . This near-ring will be
termed as right near-ring .
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Remarks :-
โข If ๐1. ๐2 + ๐3 = ๐1. ๐2 + ๐1. ๐3instead of condition (3)
the set N satisfies , then we call ๐ a left near-ring .
โข If 1. ๐ = ๐ (๐. 1 = ๐) then ๐ has a left identity(right
identity ) .
โข If (๐, +) is abelian , we call ๐ an abelian near-ring .
โข If (๐, . ) is commutative we call ๐ itself a commutative
near-ring . Clearly if ๐ is commutative near-ring then left
and right distributive law is satisfied and 1. ๐ = ๐. 1 = ๐ ,
๐ is called unital commutative near-ring .
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Example :-
Near-ring arise very naturally in the study of
mappings on groups . If (๐บ, +) is a group (not
necessarily abelian ) then the set ๐(๐บ) of all
mappings from ๐บ to ๐บ is a near-ring with respect to
pointwies addition and composition of mappings .
10
Definition :- Let ๐ be a near-ring and ๐ผ is a nonempty subset of ๐ . (๐ผ, +, . ) is called an ideal of ๐ if and only if : 1. (๐ผ, +) is normal subgroup of (๐, +) . 2. ๐ผ๐ โ ๐ผ๐ and for all ๐, ๐1 โ ๐ and for all
๐ โ ๐ผ , ๐ ๐1 + ๐ โ ๐๐1 โ ๐ผ . Definition :- An ideal ๐ of ๐ is said to be completely prime if ๐๐ โ ๐ implies ๐ โ ๐ or ๐ โ ๐ for any ๐, ๐ โ ๐ .
Let ๐ be a commutative near-ring with identity and
๐ be a completely prime ideal of ๐ . The set of all
completely prime ideal of ๐ , is denoted by ๐๐๐๐(๐) is
called near prime spectrum on completely prime ideal . Let ๐ผ
is ideal of ๐ and let ๐(๐ผ) collection of all completely prime
ideal contains ๐ผ . The collection of all ๐(๐ผ) satisfies the
axioms of closed sub sets of a topology for ๐๐๐๐(๐) called
the near Zariski topology for ๐๐๐๐(๐) . 11
Theorem 1 :- The near prime spectrum ๐๐๐๐(๐) of any commutative near-ring ๐ is a compact topological space . Corollary 2 :- Let ๐ผ be a near ideal of near-ring ๐ . Then the closed subset ๐(๐ผ) of ๐๐๐๐(๐) is a compact set .
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Theorem 3 :-
Let ๐: ๐1 โ ๐2 be a near-ring unital homomorphism between
near-rings ๐1 and ๐2 . Then ๐:๐1 โ ๐2 induces a continuous map
๐โ: ๐๐๐๐(๐2) โ ๐๐๐๐(๐1) , where ๐โ ๐ = ๐โ1(๐) for every
completely prime ideal ๐ of ๐2 .
Proof :- Let ๐2 be a completely prime ideal of ๐2. Now 12 โ ๐2,
because ๐2 is a proper ideal of ๐2, then 11 โ ๐โ1 ๐2 , since
๐ 11 = 12. It follows that ๐โ1(๐2) is a proper ideal of ๐1 . Let ๐ฅ and
๐ฆ be elements of ๐1. Suppose that ๐ฅ๐ฆ โ ๐โ1 ๐2 . Then ๐(๐ฅ)๐(๐ฆ)
= ๐(๐ฅ๐ฆ) and therefore ๐ ๐ฅ ๐ ๐ฆ โ ๐2. But ๐2 is a completely prime
ideal of ๐2, and therefore either ๐ ๐ฅ โ ๐2 or ๐ ๐ฆ โ ๐2 .Thus either
๐ฅ โ ๐โ1 ๐2 or ๐ฆ โ ๐โ1 ๐2 . This shows that ๐โ1 ๐2 is a completely
prime ideal of ๐1.
13
We conclude that there is a well-defined function
๐โ: ๐๐๐๐ ๐2 โ ๐๐๐๐ ๐1 such that ๐โ ๐2 = ๐โ1 ๐2 for all
completely prime ideal ๐2 of ๐2. Now we prove that ๐โ is a
continuous function , let ๐ผ1 be a ideal of ๐1 ,
๐โโ1 ๐ ๐ผ1 = ๐2 โ ๐๐๐๐ ๐2 : ๐โ ๐2 โ ๐ ๐ผ1 .
since ๐โ ๐2 = ๐โ1 ๐2
= ๐2 โ ๐๐๐๐ ๐2 : ๐โ1 ๐2 โ ๐ ๐ผ1
= ๐2 โ ๐๐๐๐ ๐2 : ๐ผ1 โ ๐โ1 ๐2
= ๐2 โ ๐๐๐๐ ๐2 : ๐ ๐ผ1 โ ๐2 = ๐ ๐ ๐ผ1
Thus , ๐โ: ๐๐๐๐ ๐2 โ ๐๐๐๐ ๐1 is a continuous function .
14
Theorem 4 :-
Let ๐ be a near-ring , let ๐ผ be a proper ideal of ๐ ,
and let ๐๐ผ : ๐ โ ๐/๐ผ be the corresponding quotient near-
ring homomorphism onto the quotient near-ring ๐/๐ผ . Then
the induced map ๐๐ผโ: ๐๐๐๐ ๐ ๐ผ โ ๐๐๐๐ (๐) maps
๐๐๐๐ ๐ ๐ผ homeomorphically onto the closed set ๐(๐ผ) .
Lemma 5 :-
If ๐ is a noetherian near-ring . Then ๐๐๐๐(๐) is a
noetherian space .
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Lemma 6 :- Let ๐ be a near-ring . Then the space ๐๐๐๐(๐) is a ๐0 space . Proof :- Let ๐1 , ๐2 โ ๐๐๐๐ ๐ and ๐1 โ ๐2 then ๐1 โ ๐2 or ๐2 โ ๐1 , let ๐ป ๐ = ๐1 โ ๐๐๐๐ ๐ : ๐ โ ๐1 and ๐1 โ ๐2. Then We get ๐1 โ ๐ป ๐ , ๐2 โ ๐ป ๐ . Thus ๐๐๐๐ ๐ is a ๐0 space . Lemma 7 :- Let ๐ be a near-ring . Then the space ๐๐๐๐(๐) is ๐1 if and only if ๐๐๐๐(๐) = ๐๐๐ฅ(๐) is the set of all near maximal ideal of ๐ . 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 ๐ and the topologies
are ๐ and ๐ then the bitopological space is referred to as
(๐ , ๐, ๐)
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 ๐ be a nonempty set and let ๐ โ ๐ ๐ we say that ๐
is minimal structure on ๐ if โ , ๐ โ ๐.
Let ๐ be a non-empty set, ๐ be a topology on ๐, let ๐ be a
minimal structure on ๐ then the triple (๐, ๐,๐) is called a
minimal bitopological space.
20
21
Let (๐, ๐, ๐) be a minimal bitopological space and ๐ด be a
sub set of ๐, ๐ด is called pre-open with respect to ๐ and ๐
, if ๐ด โ ๐๐๐ก๐(๐๐๐(๐ด)), where ๐๐๐(๐ด) is the closure of ๐ด
with respect to ๐ .
Let (๐, ๐, ๐) be a minimal bitopological space , and ๐ด โ ๐
, the intersection of all pre-closed sets containing ๐ด is
called pre-closure of ๐ด, and is denoted by ๐๐๐๐ โ ๐ถ๐ A .
That is ๐๐๐๐ โ ๐ถ๐ A is the smallest pre-closed set in ๐
containing ๐ด, also ๐ด is pre-closed if and only if ๐๐๐๐
โ ๐ถ๐ A = ๐ด .
22
Theorem 1 :-
Let (๐, ๐, ๐) be a minimal bitopological space, ๐ โ ๐,
๐ open with respect to ๐, ๐ and let (๐, ๐๐ , ๐๐) be a sub
space of (๐, ๐, ๐). If ๐บ be pre-open in ๐ with respect to ๐ ,
๐ . Then ๐บ โฉ ๐ is pre-open in ๐ with respect to ๐๐ , ๐๐ .
Corollary 2 :-
Let (๐,๐,๐) be a minimal bitopological space, let
(๐, ๐๐ , ๐๐) be a sub space of (๐, ๐, ๐) and let ๐open with
respect to ๐ , ๐ . If ๐บ be pre-closed in ๐ with respect to ๐ ,
๐ . Then ๐บโฉ๐ is pre-closed in ๐ with respect to ๐๐ , ๐๐ . 23
Theorem 3 :-
A minimal bitopological space (๐, ๐, ๐) is ๐๐ โ ๐๐-space if
and only if ๐๐๐๐ -closure of distinct points are distinct .
Proof :- Let ๐ฅ, ๐ฆ โ ๐, ๐ฅ โ ๐ฆ implies ๐๐๐๐ โ ๐ถ๐ ๐ฅ โ ๐๐๐
๐ โ ๐ถ๐(*๐ฆ+).
Since ๐๐๐๐ โ ๐ถ๐ ๐ฅ โ ๐๐๐
๐ โ ๐ถ๐(*๐ฆ+) there exists at least one point ๐ง,
such that ๐ง โ ๐๐๐๐ โ ๐ถ๐ ๐ฅ , but ๐ง โ ๐๐๐
๐ โ ๐ถ๐ ๐ฆ . We claim that
๐ฅ โ ๐๐๐๐ โ ๐ถ๐ ๐ฆ . For, let ๐ฅ โ ๐๐๐
๐ โ ๐ถ๐ ๐ฆ . Then ๐๐๐๐ โ ๐ถ๐ ๐ฅ
โ ๐๐๐๐ โ ๐ถ๐(*๐ฆ+) which is a contradiction that ๐ฅ โ ๐๐๐
๐ โ ๐ถ๐ ๐ฆ .
Hence ๐ฅ โ ๐\๐๐๐๐ โ ๐ถ๐ ๐ฆ but ๐๐๐
๐ โ ๐ถ๐ ๐ฆ is ๐๐๐-closed, so
๐\๐๐๐๐ โ ๐ถ๐ ๐ฆ is ๐๐๐-open which contains ๐ฅ but not ๐ฆ. It follows
that (๐, ๐, ๐) is ๐๐ โ ๐๐-space.
24
25
Conversely,
since (๐, ๐, ๐) is ๐๐ โ ๐๐-space, then for each two
distinct points ๐ฅ, ๐ฆ โ ๐, ๐ฅ โ ๐ฆ there exists ๐๐๐-open set ๐บ
such that ๐ฅ โ ๐บ, ๐ฆ โ ๐บ . ๐\๐บ is ๐๐๐-closed set which does
not contain ๐ฅ but contains ๐ฆ. By Definition , ๐๐๐๐ โ ๐ถ๐(*๐ฆ+)
is the intersection of all ๐๐๐-closed sets which contain *๐ฆ+.
Thus , ๐๐๐๐ โ ๐ถ๐(*๐ฆ+) โ ๐\๐บ, then ๐ฅ โ ๐\๐บ. This implies
that ๐ฅ โ ๐๐๐๐ โ ๐ถ๐(*๐ฆ+) , but ๐ฅ โ ๐๐๐
๐ โ ๐ถ๐(*๐ฅ+) , ๐ฅ โ ๐๐๐๐
โ ๐ถ๐(*๐ฆ+). Therefore ๐๐๐๐ โ ๐ถ๐ ๐ฅ โ ๐๐๐
๐ โ ๐ถ๐ ๐ฆ .
Theorem 4 :- Every subspace of ๐๐ โ ๐๐-
space is ๐๐ โ ๐๐-space .
Theorem 5 :- Every subspace of ๐๐ โ ๐1-
space is ๐๐ โ ๐1-space .
Theorem 6 :- Every subspace of ๐๐ โ ๐2-
space is ๐๐ โ ๐2-space .
Theorem 7 :- Every subspace of a ๐๐๐-
regular space is a ๐๐๐-regular space.
Theorem 8 :- Every ๐๐๐-closed subspace
of a ๐๐๐-normal space is a ๐๐๐-normal.
26
Theorem 9 :-
A minimal bitopological space (๐, ๐, ๐) is an ๐๐ โ ๐1-
space If and only if every singleton subset *๐ฅ+ of is ๐๐๐-
closed set.
Proof :- Suppose ๐ is an ๐๐ โ ๐1-space. Then for ๐ฅ, ๐ฆ โ ๐
and ๐ฅ โ ๐ฆ, there are two ๐๐๐-open sets ๐บ and ๐ป such that
๐ฅ โ ๐ป, ๐ฆ โ ๐ป and ๐ฆ โ ๐บ and ๐ฅ โ ๐บ. So ๐บ โ *๐ฅ+๐ also
โช *๐บ โถ ๐ฆ โ ๐ฅ+ โ *๐ฅ+๐ and *๐ฅ+๐โโช *๐บ: ๐ฆ โ ๐ฅ+ . Hence
*๐ฅ+๐=โช *๐บ: ๐ฆ โ ๐ฅ+, which is a ๐๐๐-open set. Then *๐ฅ+ is a
๐๐๐-closed . Conversely, Let ๐ฅ, ๐ฆ โ ๐ and ๐ฅ โ ๐ฆ. If *๐ฅ+ and
*๐ฆ+ are the ๐๐๐-closed sets of ๐ฅ and ๐ฆ respectively such that
๐ฅ โ *๐ฆ+, then *๐ฅ+๐ and *๐ฆ+๐ are ๐๐๐-open sets such that
๐ฆ โ *๐ฅ+๐and ๐ฅ โ *๐ฅ+๐also ๐ฅ โ *๐ฆ+๐and ๐ฆ โ *๐ฆ+๐. Then ๐ is
an ๐๐ โ ๐1-space . 27
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