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IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN:2319-765X. Volume 10, Issue 3 Ver. I (May-Jun. 2014), PP 53-61
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Topological Projective Modules of topological ring R
Dr. Taghreed Hur Majeed
Department of Mathematics- College of Education – Al-Mustansiriya University
Abstract: Our principle aim to study an algebraic module properties topolocically. We concern topological
projective modules specially. A topological module P is called topological projective module if for all
topological module epimorphism g : B  A and for all topological module morphism f :P  B, there exist
a topological module morphism f*: P  A, for which the following diagram commutes:
P
f
A
g
B
0
The important point of this paper is to find topological projective module from topological module given.
Furthermore among results are obtained at the end of the paper.
Key words: Topological module, topological ring, topological submodule, topological projective module.
I.
Introduction
Kaplanski is the first scientist who introduced the definition of topological module and topological
submodule in 1955, after that a topological module studied by many scientists likes Alberto Tolono and Nelson.
In this research a necessary and sufficient condition for topological module to be topological projective module
has been established, the kind of topological don't determinant. The result in this paper algebraically satisfied,
now we will satisfy it algebraically and topologically.
This work divided into two sections. In section one includes some necessary definitions while section two
includes propositions. We concern of this research is to obtain topological projective module from topological
module given we suppose n, n1, n2, …, nm be natural number.
1- Topological Modules
In this section we give the fundamental concepts of this work.
Definition (1.1) : [3]
A non empty set E is said to be topological group if:
(1) E is a group.
(2)  is A Topology on E.
(3) A mapping : E  E  E and : E  E are continuous, where  defined as (x) = x – 1 and E  E is
the product of two topological spaces.
Definition (1.2) : [4]
Let E be topological group. A subset M of E is said to be a topological subgroup of E if:
(1) M is a subgroup of E.
(2) M is a subspace of E.
Definition (1.3) : [1]
A non empty set R is said to be topological ring if:
(1) R is a ring.
(2)  is A Topology on R.
(3) A mapping : R  R  R defined as (x,y) = xy and : R  R defined as (x) = x
continuous.
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Topological Projective Modules of topological ring R
Definition (1.4): [1]
Let R be a topological ring. A non empty set E is said to be topological module on R, if:
1. E is module on R.
2. E is a topological group.
3. A mapping :RE  E defined as (,x) =x is continuous for all   R, x  E.
Example (1.5): [2]
Let E be a topological module on topological ring R, a subset M of E is said to be topological submodule of
E if:
1. M is a submodule of E.
2. M is a topological subgroup of E.
Proposition (1.6): [2]
Let R be topological ring and {E}1    n be finite discrete topological module of R then the direct sum
 Eα
1 α  n
be discrete topological module.
Corollary (1.7): [2]
Let R be a topological ring, {E}∆ be a family of discrete topological module then the direct sum
 Eα
α
be discrete topological module.
Definition (1.8): [3]
Let E and E be two topological modules, the map f from E into E' is said to be homomorphism topological
module if:
1. f is homomorphism module.
2. f is continuous map.
Definition (1.9): [3]
Let E and E be two topological modules on the map f from E into E is said to be homomorphism
topological module, if:
1. f is homomorphism module.
2. f is topological homeomorphism.
2- Topological Free Modules
Definition (2.1): [5]
A topological module P is said to be topological projective module if for all topological module epimorphism g:
B  A and for all topological module morphism f: P  B, there exists a topological module morphism
f *: P  A, for which the following diagram commutes:
P
f
A
g
B
Notation (2.2): [1]
Ker f is topological submodule of E, where f is topological module homomorphism from E (topological
module) into E' (topological module).
Proposition (2.3): [3]
Let P be a topological projective on a ring R and P be discrete topological module, then P is topological
projective module.
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Topological Projective Modules of topological ring R
Proposition (2.4): [3]
If
{Eα }α
be a family of topological modules on topological ring R, B be a topological module on the
same ring, the topological module homomorphism g: P  B for all  L, there exists a unique topological
module homomorphism g: B 
 Pα
α
for which the following diagram commutes:
P
g
 Pα
g
α
B
Proposition (2.5): [3]
If {Pα }α be a family of topological modules of topological ring R, then the direct sum P =
topological projective module if and only if if
{Pα }α
 {Pα } be
α
is a topological projective module.
Proposition (2.6):
If P1, P2 and P3 be topological module of topological ring R such that
f 1: P2  P1 and f 2: P3  P2
be two topological module homeomorphism, then there exist f 1∘ f 2: P3  P1 is topological module
homeomorphism.
Proof:
2:
Since f 1: P2  P1 and f 2: P3  P2 be two topological module homeomorphism, then there exist f 1∘ f
P3  P1
( f 1∘ f 2)(p3) = f 1( f 2(p3)) , for all p3  P3
= f 1(p2)
= p1
, for all p2  P2
, for all p1  P1
f 1∘ f 2 is continuous map because f 1 and f 2
are continuous maps. And ( f 1∘ f 2) – 1 is continuous map since
f 11 and f 21 are continuous maps.
Thus f 1∘ f 2 is topological module homeomorphism.
f2
f1
f 1∘ f 2
Proposition (2.7):
If P1, P2, P3, …, Pn be topological module of topological ring R such that f k: Pk + 1  Pk, 1  k  n – 1
topological module homeomorphism, then there exist i: Pn  P1 is topological module homeomorphism.
Proof:
We prove it on the same way of proposition (2.6).
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Topological Projective Modules of topological ring R
𝜄
h
f
g
B
Proposition (2.8):
If P1, P2 and P3 be topological module of topological ring R such that P1 and P2 are topological projective
modules and i : P3  P1 be topological module homeomorphism, then P3 is topological projective module
too.
Proof:
We take the following diagram to obtain
A and B are topological modules of R.
f : P1  B is topological module homeomorphism.
And g: A  B is surjective topological
module homeomorphism.
Since P1 is topological projective module, there exists
f *: P1  A is topological module homeomorphism.
Now we define h: P3  B by form
h(p3) = f *∘ i(p3) , for all p3  P3 and
g∘ h(p3) = g∘ f *∘ i(p3)
= f ∘ i(p3)
= f (p3).
Thus P3 is topological projective module.
Proposition (2.9):
If P1, P2, P3, …, Pn be topological module of topological ring R such that P1, P2, P3, …, Pn – 1 be topological
projective modules and i: Pn  P1 be topological module
homeomorphism, then Pn is topological projective module too.
Proof:
We take the following diagram to obtain
A and B are topological modules of R.
f : P1  B is topological module homeomorphism.
And g: A  B is surjective topological
module homeomorphism.
Since P1 is topological projective module, there exists f *: P1  A is
topological module homeomorphism.
Now we define h: P3  B by
h(p3) = f *∘ i(pn) , for all pn  Pn
where f1 ∘ f2 ∘…∘ fn – 2 ∘ fn – 1 = i and
g∘ h(p3) = g∘ f *∘ i(pn)
= f ∘ i(pn)
= f (pn),
for all pn  Pn
Thus Pn is topological projective module too.
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Topological Projective Modules of topological ring R
Proposition (2.10):
If P and
 Pα
1α m
be two topological modules of topological ring R such that P be topological projective
 Pα
module and i:
1α m
 P be topological module homeomorphism, then
 Pα
is topological
 Pα
be topological
1α m
projective module too.
Proof:
We take the following diagram to obtain
A and B are topological modules of R.
f : P  B is topological module homeomorphism.
And g: A  B is surjective topological
module homeomorphism.
Since P is topological projective module, there exists
f *: P  A is topological module homeomorphism.
Now we define h:
 Pα
1α m
 A by
h(q) = f *∘ i(q) , for all q 
 Pα ,
1α m
q = p1  p2  …  pm and
g∘ h(q) = g∘ f *∘ i(q)
= f ∘ i(q) = f (q).
 Pα
Thus
1α m
is topological projective module too.
Proposition (2.11):
If
 Pα
1α m
and P be two topological modules of topological ring R such that
projective module and i: P 
 Pα
1α m
1α m
be topological module homeomorphism, then P is topological
projective module too.
Proof:
We look at the following diagram to obtain
A and B are topological modules of R.
f:
 Pα
1α m
 B is topological module homeomorphism.
And g: A  B is surjective topological
module homeomorphism.
Since
f *:
 Pα
1α m
 Pα
1α m
is topological projective module, there exists
 A is topological module homeomorphism.
Now we define h: P  A by
h(p) = f *∘ i(p) , for all p  P and
g∘ h(p) = g∘ f *∘ i(p) = f ∘ i(p) = f (p) , for all p  P.
Thus P is topological projective module too.
Proposition (2.12):
If
 Pα1
1α1  n1
and
 Pα2
1α 2 n 2
topological projective module and i:
 Pα2
1α 2 n 2
be two topological modules of topological ring R such that
 Pα2

1α 2 n 2
 Pα1
1α1  n1
 Pα1
1α1  n1
be
be topological module homeomorphism, then
is topological projective module too.
Proof:
We look at the following diagram to obtain
A and B are topological modules of R.
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Topological Projective Modules of topological ring R
f:
 Pα1
1α1  n1
 B is topological module homeomorphism.
And g: A  B is surjective topological
module homeomorphism.
 Pα1
Since
f *:
1α1  n1
 Pα1
1α1  n1
is topological projective module, there exists
 A is topological module homeomorphism.
Now we define h:
 Pα2
1α 2 n 2
 A by
 Pα2 ,
h(q) = f *∘ i(q) , for all q 
1α 2 n 2
q  p1  p2  ...  pn 2
where
and
g∘ h(q) = g∘ f *∘ i(q)
= f ∘ i(q)
= f (q).
 Pα2
Thus
is topological projective module too.
1α 2 n 2
Proposition (2.13):
 Pα1 ,
If
 Pα2
1α1  n1
 Pα1
1α1  n1
1α 2 n 2
 Pα2
and
1α 2 n 2
and
 Pα3
be topological modules of topological ring R such that
1α3  n 3
are topological projective modules and
topological module homeomorphism, then
 Pα3
1α3  n 3
i:
 Pα3
1α3  n 3

 Pα1
1α1  n1
be
is topological projective module too.
Proof:
We look at the following diagram to obtain A and B are topological modules of R.
f:
 Pα1
1α1  n1
 B is topological module homeomorphism.
And g: A  B is surjective topological module homeomorphism.
 Pα1
Since
f *:
1α1  n1
 Pα1
1α1  n1
is topological projective module, there exists
 A is topological module homeomorphism.
Now we define h:
 Pα3
1α3  n 3
h(q) = f *∘ i(q) , for all q 
where
 A by
 Pα3 ,
1α3  n 3
q  p1  p2  ...  pn3
and
g∘ h(q) = g∘ f *∘ i(q)
= f ∘ i(q)
= f (q).
Thus
 Pα3
1α3  n 3
is topological projective module.
Proposition (2.14):
If
 Pα1 ,
1α1  n1
such that
 Pα2 ,  Pα3 , …,
1α 2 n 2
1α3  n 3
 Pα1 ,  Pα2 ,  Pα3 , …,
1α1  n1
1α 2 n 2
1α3  n 3
 Pαm
1α m  n m

be topological modules of topological ring R
Pαm1
are topological projective modules and i:
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1α m 1  n m 1
Topological Projective Modules of topological ring R
 Pαm

1α m  n m
 Pα1
be topological module homeomorphism, then
1α1  n1
 Pαm
1α m  n m
is topological
projective module too.
Proof:
We look at the following diagram to obtain
A and B are topological modules of R.
f:
 Pα1
1α1  n1
 B is topological module homeomorphism.
And g: A  B is surjective topological
Since
f *:
 Pα1
1α1  n1
 Pα1
1α1  n1
is topological projective module, there exists
 A is topological module homeomorphism.
Now we define h:
 Pαm
1α m  n m
h(q) = f *∘ i(q) , for all q 
 A by
 Pαm ,
1α m  n m
q  p1  p2  ...  pnm
where
and
g∘ h(q) = g∘ f *∘ i(q)
= f ∘ i(q)
= f (q).
Thus g∘ h = f and
 Pαm
1α m  n m
Proposition (2.15):
 {Pα1 }
If
α1
and
is topological projective module.
 {Pα2 } ,
α 2
be a families of topological modules of topological ring R such that
 {Pα1 } be topological projective module and
α1
topological module homeomorphism, then
i:
 {Pα2 }
α 2

 {Pα1 } be
α1
 {Pα2 } is topological projective module too.
α 2
Proof:
We look at the following diagram to obtain
A and B are topological modules of R.
f:
 {Pα1 }  B is topological module homeomorphism.
α1
And g: A  B is surjective topological
module homeomorphism.
Since
f *:
 {Pα1 } is topological projective module, there exists
α1
 {Pα1 }  A is topological module homeomorphism.
α1
Now we define h:
 {Pα2 }  A by
α 2
h(q) = f *∘ i(q) , for all q 
where
 {Pα2 } ,
α 2
q  p1  p2  ...  pnm
and
g∘ h(q) = g∘ f *∘ i(q)
= f ∘ i(q)
= f (q).
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Topological Projective Modules of topological ring R
Thus g∘ h = f and
 {Pα2 } is topological projective module.
α 2
Proposition (2.16):
If
 {Pα1 },  {Pα2 }
α1
such that
α 2
and
 {Pα3 } , be a families of topological modules of topological ring R
α3
 {Pα1 } and  {Pα2 } are topological projective modules and i:  {Pα3 }   {Pα1 } be
α1
α3
α 2
topological module homeomorphism, then
α1
 {Pα3 } is topological projective module too.
α3
Proof:
We look at the following diagram to obtain A and B are topological modules of R.
f:
 {Pα1 }  B is topological module homeomorphism.
α1
And g: A  B is surjective topological
module homeomorphism.
Since
 {Pα1 } is topological projective module,
α1
there exists f *:
 {Pα1 }  A
α1
is topological module homeomorphism.
Now we define h:
 {Pα3 }  A by
α3
h(q) = f *∘ i(q) , for all q 
 {Pα3 } and
α3
g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q).
Thus g∘ h = f and
 {Pα3 } is topological projective module.
α3
Proposition (2.17):
 {Pα1 } ,  {Pα2 } ,  {Pα3 } ,
If
α1
α3
α 2
topological ring R such that
projective modules and i:
…,
 {Pαm }
α m 
be a families of topological modules of
 {Pα1 } ,  {Pα2 } ,  {Pα3 } ,
α1
 {Pαm }
α m 
α 2

α3
 {Pα1 }
α1
…,
 {Pαm 1 }
αm 1
are topological
be topological module homeomorphism, then
 {Pαm } is topological projective module too.
α m 
Proof:
We look at the following diagram to obtain
A and B are topological modules of R.
f:
 {Pα1 }  B is topological module homeomorphism.
α1
And g: A  B is surjective topological
module homeomorphism.
Since
f *:
 {Pα1 } is topological projective module, there exists
α1
 {Pα1 }  A is topological module homeomorphism.
α1
Now we define h:
 {Pαm }  A by
α m 
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Topological Projective Modules of topological ring R
h(q) = f *∘ i(q) , for all q 
 {Pαm } and
α m 
g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q).
Thus g∘ h = f and
 {Pαm } is topological projective module.
α m 
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