Z03802480262

American Journal of Engineering Research (AJER) 2014 American Journal of Engineering Research (AJER) e-ISSN : 2320-0847 p-ISSN : 2320-0936 Volume-03, Issue-08, pp-248-262 www.ajer.org Research Paper

Open Access

Insight on the Projective Module Udoye Adaobi Mmachukwu¹ Akoh David² ¹ Department of Mathematics, Federal University Oye-Ekiti, Ekiti State, Nigeria. ² Department of Mathematics, Federal Polytechnic, Bida, Niger State, Nigeria.

Abstract : In this work, we focus on some important results on modules and projective modules. We discuss some aspects of modules laying emphasis on its axioms, exactness and free module.

Keywords: submodules, exactness, split exactness, homomorphism

1. Introduction The theory of module is an important branch in Algebra. There are many applications of projective module as studied by some researchers. Yengui (2011) observed that for any finitedimensional ring, all finitely generated stably free modules over

of rank

dim

are

free; the result was only known for Noetherian rings. According to T t nc et al. (2012) if is a semi-projective End

-coretractable module with finite hollow dimension and

, there exist

such that

hollow submodule of

and

, where for each

is a nonzero . The

paper discusses concept of module, exactness of sequence. There are two classes of modules: left and right -modules. The result on left on right -modules. Since

-

-modules yields a corresponding result

is commutative, both left -module and right

-module will be

regarded as -module.

Definition 1.1. Let operation “+” on

be a ring with identity. A left under which

-module is a set together with a binary

is an abelian group, and an action on

, that is, a map

, which satisfies the following axioms: (a.) (b.) www.ajer.org

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American Journal of Engineering Research (AJER) 2014 (c.) (d.)

.

Similarly,

a

right

-module

is

an

abelian

group

together

with

a

map

, satisfying the following axioms: For (a.) (b.) (c.) (d.)

.

Every abelian group is a - module.

1.2. Submodule. Let Submodule of

if

be an -module. A non-empty subset is an additive subgroup of

by elements of , as for by

of

is called a

closed under the “same� multiplication

. This implies that the axioms of an -modules must be satisfied

.

2. -Homomorphisms between two modules Let

and

be two -modules, then a function and all

is a homomorphism in case . The function

the defining structure in order to be a module homomorphism. Precisely, if are -modules. A mapping

must preserve is a ring and

is called an -module homomorphism if

(i)

.

(ii)

.

2.1. Epimorphisms and Monomorphisms A homomorphism

is called an epimorphism in case it is surjective. That is, if

maps the elements of module

onto

. An injective -homomorphism

called an -monomorphism (that is, one-to-one). An -homomorphism isomorphism in case it is a bijection. Thus, two modules

and

denoted by

. An -homomorphism

, if there is an -isomorphism is called an

-endomorphism. If

is is an -

are said to be isomorphic,

is bijective, then, it is called an

automorphism.

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-

American Journal of Engineering Research (AJER) 2014 2.1.1 Kernel and Image of R-homomorphism. Let modules and

be a ring. Let

be an - homomorphism. The Kernel of

by

and

be

denoted by

-

is given

.

The image of

denoted by

or

is defined as

Proposition 2.1.2.

Let

and

be -modules and

(i)

is a submodule of

(ii)

is a submodule of .

be an -module homomorphism. .

(iii)The -homomorphism Proof: (i) Let

is a monomorphism if and only if

. We claim that

and

by the definition of

This implies that Again, let

. Let

and

homomorphism. Since (ii) Let

. ⇒

. is a module,

. Hence,

since Thus,

.

is a submodule of

is a .

such that

is a module). Therefore, such that

is arbitrary.

since

. Then, there exists

exists

since

. Then

is a homomorphism.

since (since

be a zero of

.

since

.

;

is a homomorphism. But . Also, let

.

, since

is a module. Therefore,

and

. There

is a homomorphism. But

.

is a submodule of .

(iii) If: Let

and

for

. ⇒

⇒

⇒

is a monomorphism.

Only if: Let

be a monomorphism and

monomorphism and

,

. Then,

since

is a

.

2.2. Factor Module

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American Journal of Engineering Research (AJER) 2014 Theorem and definition: Let . Then

be an

-module. Let

is an

-module, where addition in

for any scalars in Proof:

be a submodule of

is defined as follows: for

and

is defined by

and closure under multiplication by

;

,

.

is an additive abelian group. Also, closure by scalars in

definition. Now, to verify module axioms. Generally, for

,

follows from . Let

.

(i) . (ii) . (iii)

(by associativity in ) .

If

has identity

, then,

.

2.3 Exactness 2.3.1 Exact Sequence. Let

and

is said to be exact at if and only if there exists

. The pair of in case

homomorphisms

. That is, let

such that

, then

. In general, a finite or

infinite sequence of homomorphism

is exact

in case it is exact for each successive pair

and

.⇒

consecutive terms, the subsequence

. ⇒ for any three

is exact.

2.3.2 Short exact Sequence An exact sequence of the form

is called a short exact

sequence.

Proposition 2.3.3. If the sequence sequence. Then,

is a monomorphism and

Proof. The exactness of injective. Similarly,

is a short exact is an epimorphism. means that

is surjective is equivalent to

Thus, in the given short exact sequence is surjective and www.ajer.org

. ⇒

is a monomorphism and

. . This implies that is exact. ,

is injective,

is an epimorphism.

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is

American Journal of Engineering Research (AJER) 2014 Remark. Let

and

be any two -modules. Every -homomorphism from

element of the set of functions from notation;

to

to

is an

. These homomorphisms form a set with standard

.

Theorem 2.3.4.

is an -module, where

is a commutative ring and

are

-modules.

Proof: Suppose

is commutative. For . Since

,

,

and

, define

by

, .

We have that

. Also, let

scalar multiplication. Let

and

and

preserves the

be arbitrarily given. Then, we have; .

Hence,

preserves scalar multiplication. Since and

, for every

we define a function . Thus,

Remark. From the above theorem, if is also an

is commutative, for arbitrary given by taking â–Ą

is an -module.

is a commutative ring and

-module, where

is an

is considered an

is a module called the dual of

and it is denoted by

-module, then,

-module over itself.

.

2.4 Direct Summands, Split Homomorphism Suppose

and

are submodules of a module

.

and

.

Then,

module with

is epic if and only if

independent. If

direct sum of its submodules

if

in case

from the

,

and

and

. Let

-homomorphism

span

, and monic if and only if

$ the canonical homomorphism is an isomorphism, then

, there exists unique

span

are linearly independent in case be a canonical

cartesian product

and

, and it is denoted by and

such that

. and

are

is the (internal) . Thus, for each if and only

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American Journal of Engineering Research (AJER) 2014 A submodule

of

that

is a direct summand of

.

in case there is a submodule

is also a direct summand.

direct summands. More generally, Let and each

and

such

are called Complementary

be submodules of an -module

has a unique representation

,

’s and it is written as

called the internal direct sum of

of

. Suppose

. Then,

is

.

2.4.1 Split exact Sequence Let

and

be homomorphisms with

write

and we say

. Thus, a

is split or is split exact in case

is a

is a split epi.

Lemma 2.4.2. Let

and

is an epimorphism, Proof:

is a split epi and we

is a split mono, and write

short exact sequence split mono and

, then

Clearly,

be homomorphisms such that

is a monomorphism, and is

epi,

,

is

monic.

and

. Then,

.

If

then .

If

,

then

and .⇒

.

□

Proposition 2.4.3. The following statements about a short exact sequence

are equivalent in

.

(a.) The sequence is split; (b.) The monomorphism (c.) The epimorphism (d.)

is split; is split;

is a direct summand of

;

(e.) Every homo

factors through ;

(f.) Every homo

factors through .

3.1 Free Module Definition 3.1.1.

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American Journal of Engineering Research (AJER) 2014 Let

be a non-empty subset of an -module

or base for

if every element

elements of . That is,

.

is called a set of free generators or basis

can be uniquely expressed as a linear combination of uniquely where

only a finite number of

ranges through the elements of

's are non-zero in . A subset

of a module

over

and

is said to be

linearly independent if and only if for every finite number of distinct elements

,

n



 i xi

,⇒

,

. An -module

is free if it has a basis. A non-

i 1

zero cyclic module basis for

if

and only if

where each

Theorem 3.1.2. Given any -module

Then,

of

. Define a mapping ,

with

. If

Theorem 3.1.4. Let

is surjective since

is said to be finitely generated (FG) if

and

can be generated

is finitely generated if there is a

are finitely generated, so is

be a basis for a free

-module,

and

Proof: Let

, then,

number of the

's are non-zero. Define a map

finite

.

any -module. Let

be any mapping. Then, there exists a unique -homomorphism,

.

by

is exact. □

. Then,

by some finite set of elements. Thus, a module subset

if

is a free -module.

-module with basis

Definition 3.1.3. An -module

is a basis for

.

extends to a unique homomorphism

. Let

is a

, there exists an exact sequence

such that be a free

. Precisely,

is a free cyclic module. Thus,

is linearly independent and generates

Proof: Let

⇒

is said to be free if given

can be expressed in the form

such that

.

where only a finite by

. Thus,

□

Remark. A homomorphism to factor through

that is composite of homomorphisms

and . A homomorphism

whose kernel is contained in that of

is said

factors uniquely through every epimorphism

and through every monomorphism whose image

contains the image of .

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American Journal of Engineering Research (AJER) 2014 3.2 Projective modules Projection. Let

be an

-module. Let

direct summand

, that is,

be a direct summand of . Then,

with complementary

, where

defines an epimorphism

called the projection of

Proposition 3.2.1. Let

as given above, then, the projection of

is the unique epimorphism Proof:

of

then,

then

the

of

,

on

on

along

and

all

as its complementary direct summand. If projection

.

.

be such that

for

with complementary direct summand with

along

which satisfies

satisfies the given conditions. Let ,

on

,

,

,

. Again, if

is the direct summand

. Then,

is a direct summand of

is the projection of

along

and

can

be

on

along

characterized

, by;

.

Theorem 3.2.2. Suppose that

and

represent inclusion mapping where

, then

are split exact.

Proposition 3.2.3 Let be a submodule of

. Let

. Then,

be the projection of if and only if

Projective modules: A module

on

along

, and let

is an isomorphism.

is called a Projective module if given any diagram M

h

g

P

N f

where

is surjective and

homomorphism

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is an

-module homomorphism, there exists a

such that the diagram is commutative.

is a lifting of . In other

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American Journal of Engineering Research (AJER) 2014 words, given an epimorphism as

, then, any homomorphism

can be factored

( is lifted to ).

Theorem 3.2.4. A free module is projective.

Proof: Suppose that the -module be any

-homomorphism. Since

and for each by

is free. Let

is free, it has a basis. Let

, there exists

such that

. Define

Theorem 3.2.5. Every direct summand of a projective module over Proof:

Let

be a projective module with be a projection map,

is projective.

, a direct summand of

. Let

be an epimorphism,

homomorphism; where

be any

-

is an inclusion map.

,

⇒

. Then,

⇒

is a

is projective. □

⇒

lifting to

be a basis for .

is projective. □

. ⇒

. Then,

be an -epimorphism and

Theorem 3.2.6. Let

be an

-module. Then,

is projective if and only if

is a direct

summand of a free -module.

Proof: Assume

is projective and

Let

is a free module, there exists an exact sequence

be identity map. Then, there exists

is a split short exact sequence. ⇒ Conversely, any

let

be the projection map,

-homomorphism, then,

free and is projective. ⇒

. lifts to

Let

. an

lifts to

. ⇒

. ⇒

such that

-epimorphism and since

is

be the inclusion map. Then,

is projective. □

Theorem 3.2.7. A direct sum of projective modules is projective if and only if each summand is projective.

Proof:

Let

be the direct sum of

-module

,

. Suppose each

projective and consider the diagram M

www.ajer.org A

M

0

Page 256

is

American Journal of Engineering Research (AJER) 2014

where the row is exact. Let

be the restriction of

there exists ⇒

with



to

. By projectivity of

. Let

,

. Then,

is projective.

Conversely, assume

is projective and we have the diagram with exact row, M

A

extend



to

0

B



by defining

. Then the restriction

Theorem 3.2.8. An

-module

. Let

for some make the diagram to be

⇒ each summand of

commutative. Thus,



is projective. □

is projective if and only if every short exact sequence splits.

Proof: Assume

is projective. Let

be exact. Given the diagram below P

N

P g

By hypothesis, we can fill in the diagram with Thus,

and the short exact sequence

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to obtain a commutative diagram. splits.

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American Journal of Engineering Research (AJER) 2014 Conversely, suppose the short exact sequence is a direct summand of Theorem 3.2.9. Let

splits, then,

is projective.

and hence,

□

be a commutative ring with identity;

and

,

-modules. Suppose

, then (a.)

given by

(ii)

if

is an

-homomorphism.

is defined.

(b.) Let

be an exact sequence. Then, is exact.

Proof:

(a.)

Let

be

a

homomorphism.

given by

is an

, then for each , there exists -homomorphism.

To

show

that

-homomorphism. Let . We claim that

is an

. For and

and for

all

,

⇒ (ii.)

is an -homomorphism. is epic if and only if for each

that

. Let

. Since

where

, there exists

exact.

⇒

is an -homomorphism,

.

(b.) Given that

is

such

is exact. To show that

Let

. ⇒

There

exists

such

that

is surjective. Thus, the exactness of is established. □

Theorem 3.2.10.

An

-module

is projective if and only if for any exact sequence of -modules, the sequence is exact.

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American Journal of Engineering Research (AJER) 2014 Proof: Let

be exact. Suppose

given

,

there

is projective, then

exists

. In this case,

such

that

is surjective and we have the

exactness of .

The converse holds. Suppose

is exact and

(with

). Let

. Then, we have the exact sequence the injection map.

where

Applying the exactness of

we have that

is

is projective.

â–Ą

Conclusion.

A module

is finitely generated and projective if and only if

module with a finite base since if then

is a direct summand of a free module

is projective and is also a homomorphic image of a free module

finite set of generator. Also,

with a finite base, . Thus,

has a

being a finitely generated and projective module implies that

we have an exact sequence base.

is a direct summand of a free

is projective implies that

where is isomorphic to

, thus,

is free and has a finite

is a direct summand of a

free module with finite base.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Anderson, W.F., Graduate text in Mathematics: Rings, and Categories of Modules, Springer-Verlag, New york, 1992.

Dummit, S. and Foote, M., Abstract Algebra, University of Vermont, 1991.

Eckmann, B., Moser J.K. and Stenstrom, B., ``Rings of Quotients: An introduction to Methods of Rings '', Springer-Verlag, New york, 1975.

Hartley, B. and Hawkes ,T.O., Rings, Modules and Linear Algebra, Chapman And Hall Ltd., London, 1970.

Ilori, S.A , Lecture Note on MAT 710, University of Ibadan, 2008.

Jacobson ,N., Basic Algebra II, Revised Edition, Vol.2, W.H Freeman and Company, 1989.

Kaplansky , I., Chicago Lectures in Mathematics: Field and Rings, Second Edition, University of Chicago, 1972.

Kuku, A. O., Abstract Algebra, Ibadan University Press, Ibadan, 1992.

Maclane S.and Birkhoff, G., Algebra, Third Edition, New York, 1965.

Reis, C., Ring Theory, Academic Press, New York, 1972.

Silvester, J.R., Introduction to Algebra K-theory, Chapman and Hall, Great Britain, 1981.

Sze-Tsen, H., Elements of Modern Algebra, University of Carlifornia, Los Angeles, 1965.

www.ajer.org

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American Journal of Engineering Research (AJER) 2014 [13]

[14]

, D.K., Toksoy, S.E. and Tribak,R., A Note on Endomorphism Rings of Semi-Projective Modules, Mathematical Proceedings of the Royal Irish Academy, Vol. 112A, No. 2, (pp. 93-99), 2012.

Yengui, I., Stably Free Modules over of Rank dim are free, Mathematics of Computation. Vol. 80, No. 274, (pp. 1093-1098), 2011.

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