ON FUZZY DIMENSION OF A MODULE WITH DCC ON SUBMODULES

Acharya Nagarjuna International Journal of Mathematics & Information Technology Vol.1, No.1, pp 13-32

ANIJMIT Acharya Nagarjuna

University - 2004 ON FUZZY DIMENSION OF A MODULE WITH

DCC ON SUBMODULES

Satyanarayana Bhavanari Department of Mathematics, Acharya Nagarjuna University,

Nagarjuna Nagar – 522 510, Andhra Pradesh, INDIA.

Godloza Lungisile Department of Mathematics, University of Transkei,

Private Bag XI, Umtata, SOUTH AFRICA.

and

Mohiddin Shaw Shaik Department of Mathematics, ANU PG Centre at Ongole

Ongole – 523 001, Andhra Pradesh, INDIA.

(Communicated by Prof. Dr. P.V. Arunachalam)

Abstract

Let R be fixed (not necessarily commutative) ring with identity, and M be a unitary left R-module. Goldie [ 7 ] introduced the concept of Finite Goldie dimension (FGD, in short) in modules. Later this dimension theory was studied and developed by the authors like: Anh & Marki [ 1, 2 ]; Reddy & Satyanarayana [ 11 ]; Satyanarayana [ 12 ]; and Satyanarayana, Syam Prasad & Nagaraju [ 19 ]. After the introduction of Fuzzy set by Zadeh [ 21 ], the researchers in Mathematics were trying to introduce and study this concept of Fuzzyness in different mathematical systems under study. Fu-Zheng Pan [ 4, 5 ]; and Golan [ 6 ] were studied the concept ‘fuzzy submodule’. In the present paper, we introduced the concepts: minimal element, fuzzy linearly independent element, fuzzy basis, fuzzy dimension in modules, and proved some important theorems (Th. 3.10, Th. 4.2 and Th. 5.5) related to these concepts.

Key words: Finite Goldie Dimension, uniform element, u-linearly independent element, fuzzy submodule, fuzzy dimension. AMS (2000) Subject Classification: 16D 10, 16P 60, 03E 72.

14 Satyanarayana, Godloza, Mohiddin

1. Introduction

It is well known that the dimension of a vector space is defined as

the number of elements in the basis. One can define a basis of a vector

space as a maximal set of linearly independent vectors or a minimal set

of vectors which span the space. The former, when generalized to

modules over rings, becomes the concept of Goldie Dimension. The

concept of Goldie Dimension in modules was studied by several authors

like Anh, Marki, Goldie, Reddy, Satyanarayana, Syam Prasad and

Nagaraju.

Now we collect the necessary literature. Let R be a fixed (not

necessarily commutative) ring. Throughout this paper, we are concerned

with left R-modules M. Like in Goldie [ 7 ], we shall use the following

terminology: A non-zero submodule K of M is called essential in M (or

M is an essential extension of K) if K ∩ A = (0) for any other

submodule A of M, implies A = (0). If K is essential in M then we

denote this fact by K ≤e M. It is easy to verify that the intersection of

finite number of essential submodules is essential. If I, J, K are

submodules of M such that I ≤e J, and J ≤e K, then I ≤e K. Also if

I ⊆ J, then I ≤e J ⇒ (I ∩ K) ≤e (J ∩ K). It is also true that if the sum of

modules {Gi / 1 ≤ i ≤ n} is direct and Hi is a submodule of Gi for

1 ≤ i ≤ n, then “Hi ≤e Gi, 1 ≤ i ≤ n ⇔ H1 ⊕ H2 ⊕ …⊕ Hn ≤e G1 ⊕ G2 ⊕

…⊕ Gn” is true. M has Finite Goldie Dimension (abbr. FGD) if M does

not contain a direct sum of infinite number of non-zero submodules.

Fuzzy Dimension in Modules 15

Equivalently, M has FGD if for any strictly increasing sequence

H0 ⊂ H1 ⊂ H2 ⊂… of submodules of M, there is an integer i such that Hk

is essential submodule in Hk+1 for every k ≥ i. A non-zero submodule H

of M is uniform if every non-zero submodule of H is essential in H.

Then it is proved (cf. [ 7 ]) that in any module M with FGD, there exist

uniform submodules U1, U2, …, Un whose sum is direct and essential in

M. The number n is independent of the uniform submodules. This

number n is called the Goldie dimension of M and is denoted by

dim M. It can be easily proved that if M has FGD, then every

submodule K of M has also FGD and dim K ≤ dim M. Furthermore, if

K, A are submodules of M, and K is maximal with respect to the

property that K ∩ A = (0), then we say that K is a complement of A (or a

complement in M). It is easy to prove that if K is a submodule of M,

then K is a complement in M ⇔ there exists a submodule A in M such

that A ∩ K = (0) and K1 ∩ A ≠ (0) for any submodule K1 of M such that

K1 is properly containing K. In this case, we have that K + A is essential

in M. It is proved that (Goldie [ 7 ] and Reddy & Satyanarayana [ 11 ])

if M has FGD, then a submodule K is a complement ⇔ M/K has FGD

and dim(M/K) = dim M - dim K. Satyanarayana, Syam Prasad and

Nagaraju [ 19 ] proved that if M has FGD and K1, K2 are two

submodules of M such that K = K1 ∩ K2 is a complement, then

dim K1 + dim K2 = dim (K1 + K2) + dim (K1 ∩ K2). The elements

a1, a2,…,an ∈ M are said to be linearly independent (l. i, in short) if the

sum of the submodules generated by ai’s (for 1 ≤ i ≤ n) is direct. In this

case, we also say that {ai / 1 ≤ i ≤ n} is a linearly independent set


(l. i. set, in short). The submodule generated by an element x ∈ M is

denoted by (x) or <x>. A non-zero element a ∈ M is said to be uniform

element (or u-element) if (a) is an uniform submodule. The u-elements

a1, a2, …, an are said to be u-linearly independent (u. l. i) elements if the

set {ai / 1 ≤ i ≤ n} is a l. i set. In this case, we also say that

{ai / 1 ≤ i ≤ n} is an u-linearly independent set (or u. l. i. set). If M has

FGD, n = dim M, and a1, a2, …, an are l. i elements, then we say that

{a1, a2, …, an} is a basis for M. Satyanarayana [ 14 ] proved that (i) if M

has FGD and ai, 1 ≤ i ≤ n are l. i elements, then n ≤ dim M; (ii) If M has

FGD, then “dim M” is the largest positive integer n such that there exist

l. i elements a1, a2, …,an in M. Moreover, if n = dim M and a1, a2, …,an

are l. i.elements, then each (ai), 1 ≤ i ≤ n, is uniform and the set

{ai / 1 ≤ i ≤ n} is an u. l. i set.

Next we collect necessary information related to fuzzyness from the

existing literature.

An important branch of Mathematics which deals with sets is

"Fuzzy Set Theory", used in artificial intelligence. Success of fuzzy

logic in a wide range application inspired much interest in fuzzy logic

among Mathematicians. Lotfi. A. Zadeh (a professor in Electrical

Engineering and Computer Science at University of California,

Berkeley) (July 1964) introduced a theory whose objects called ‘fuzzy

sets’ (are sets with boundaries that are not precise) (cf.[ 21 ]). In a

narrow sense fuzzy logic refers to a logical system that generalizes


classical two-valued logic for reasoning under uncertainty. Prof. Zadeh

believed that all real world problems could be solved with more efficient

and analytic methods by using the concept fuzzy sets. The fuzzy boom

(1987 to present) in Japan was a result of the close collaboration and

technology transfer between Universities and Industries. In 1988, the

Japanese Government launched a careful feasibility study about

establishing national research projects on fuzzy logic involving both

Universities and Industries. As a result the Japan is able to manufacture

fuzzy vacuum cleaner, fuzzy rice cookers, fuzzy refrigerators, fuzzy

washing machines, and others.

Let U be an Universal set. Each element of the universal set has

a degree of membership (which is a real number between 0 and 1,

including 0 and 1), in a fuzzy set S. The fuzzy set S is denoted by listing

the elements of universal set with their degrees of membership.

Normally elements with membership 0 may not be listed.

1.1 Definition: Let U be universal set. Consider U × [0, 1].

(i). Let µ: U → [0, 1] be a function. Now {(x, µ(x)) / x ∈ U} (which is a

subset of U × [0, 1]) is called a fuzzy set (or fuzzy subset of U).

(ii). A fuzzy subset µ is said to a non-empty fuzzy subset if there exists

x ∈ U such that µ(x) ≠ 0.

(iii).Let t ∈ [0, 1]. The set {x ∈ U / µ(x) ≥ t}is called a level set (or level

subset) of µ.


An example is given under to illustrate these concepts.

1.2 Example: Let U = {x, y, z, 2, 3, 4} and

S = {(x, 0.1), (y, 0.2), (z, 0.3), (4, 1), (2, 0), (3, 0)} is a fuzzy set on U.

Here µ(x) = 0.1, µ(y) = 0.2 and so on.

Note that µ0.25 = {a ∈ U / µ(a) ≥ 0.25} = {z, 4}, and µ0 = U.

Here (x, 0.1) ∈ S. In this case we say that "the degree of membership

of x in S is 0.1". Also (2, 0) ∈ S means that the degree of membership of

2 in S is 0. Therefore 2 is not in S (or 2 ∉ S).

The researchers in mathematics were trying to introduce and

study the concept of fuzzyness in different mathematical systems under

study. Negoita and Ralescu [ 10 ] introduced the concept "fuzzy

submodule" in 1975. Later this concept was studied by the authors like:

Fu-Zheng PAN [ 4, 5 ]; and Golan [ 6 ].

In this paper, we consider modules over an associative ring R

with identity. Now we present two existing definitions of fuzzy

submodules and some related fundamental results.

1.3 Definition: (Negoeta & Ralescu [ 10 ]) Let R be a ring and M a

module; and µ: M → [0, 1] a mapping. The pair ( M, µ) is said to be a

fuzzy submodule if it satisfies the following four conditions:

(i). µ(m + m1) ≥ min{µ(m), µ(m1)} for all m, m1 ∈ M;

(ii). µ(- m) = µ(m) for all m ∈ M; (iii). µ(0) = 1; and


(iv). µ(am) ≥ µ(m) for all m ∈ M and a ∈ R. We also denote the fuzzy

submodule (M, µ) by µM (in short).

Fu-Zheng PAN [ 4, 5 ] studied this concept of "fuzzy

submodules" and developed the theory. Golan [ 6 ] introduced fuzzy

submodules over a ring with identity.

1.4 Definition (Golan [ 6 ]): Let R be a ring with unit element 1. A pair

(M, µ) consisting of an unitary R-module M and a function

µ: M → [0, 1] satisfying the following three conditions: (i). µ(0) = 1;

(ii). µ(m + m1) ≥ min{µ(m), µ(m1)} for all m, m1 ∈ M.; and

(iii). µ(am) ≥ µ(m) for all m ∈ M and for all a ∈ R, is called a "fuzzy

submodule" of M.

Taking a = -1 in condition (iii) of the above definition, we get that

µ(-m) = µ(m) for all m ∈ M. From this fact, it follows that the above

two definitions for fuzzy submodule are equivalent.

2. Fuzzy Submodules

We start this section by defining the concept "fuzzy submodule",

which is a generalization of the existing definitions (1.3 & 1.4).

Throughout this paper M stands for a unitary R-module.


2.1 Definition: Let M be a unitary R-module and µ : M → [0, 1] be a

mapping. µ is said to be a fuzzy submodule if the following two

conditions hold:

(i) µ(m + m1) ≥ min{µ(m), µ(m1)} for all m and m1 ∈ M; and

(ii) µ(am) ≥ µ(m) for all m ∈ M, a ∈ R.

2.2 Proposition: If M is a unitary R-module, µ : M → [0, 1] is a fuzzy

set with µ(am) ≥ µ(m) for all m ∈ M, a ∈ R then the following two

conditions are true.

(i). for all 0 ≠ a ∈ R, µ(am) = µ(m) if a is left invertible; and

(ii). µ(-m) = µ(m).

Proof: (i) Let b be a left inverse of a. Then ba = 1. Now

µ(am) ≥ µ(m). µ(m) = µ(1.m) = µ(bam) ≥ µ(am) (since µ is fuzzy

submodule) ⇒ µ(m) ≥ µ(am). Hence µ(am) = µ(m) for all m ∈ M,

and for all left invertible elements 0 ≠ a ∈ R.

(ii) follows from (i) by taking a = -1.

2.3 Corollary: If µ: M → [0, 1] is a fuzzy submodule and m, m1 ∈ M,

then µ(m - m1) ≥ min{µ(m), µ(m1)}.

Proof: Given µ is a fuzzy submodule. Now µ(m - m1) = µ(m + (-m1))

≥ min{µ(m), µ(-m1)} (since µ is a fuzzy submodule) ≥ min{µ(m),

µ(m1)} (by the Proposition 2.2). Therefore µ(m - m1) ≥ min{µ(m),

µ(m1)} for all m, m1 ∈ M.


2.4 Proposition: If µ: M → [0, 1] is a fuzzy submodule, m, m1 ∈ M and

µ(m) > µ(m1), then µ(m + m1) = µ(m1). In other words, if µ(m) ≠ µ(m1),

then µ(m + m1) = min{µ(m), µ(m1)}.

Proof: Proof is similar to that of Proposition 1 of Golan [ 6 ].

2.5 Corollary: If µ: M → [0, 1] is a mapping satisfies the condition

µ(am) ≥ µ(m) for all m ∈ M and a ∈ R, then the following two

conditions are equivalent:

(i) µ(m - m1) ≥ min{µ(m), µ(m1)}; and

(ii) µ(m + m1) ≥ min{µ(m), µ(m1)}.

Proof: (i) ⇒ (ii): Suppose (i). Now µ(m + m1) = µ(m - (-m1)) (since M

is a module) ≥ min{µ(m), µ(-m1)} (by supposition) = min{µ(m),

µ(m1)} ⇒ µ(m + m1) ≥ min{µ(m), µ(m1)}.

(ii) ⇒ (i): follows from Corollary 2.3.

2.6 Proposition: If µ: M → [0, 1] is a fuzzy submodule, then

(i) µ(0) ≥ µ(m) for all m ∈ M ; and (ii) µ(0) = ( )mSupMmµ

∈.

Proof: (i) µ(0) = µ (x - x) ≥ min{µ(x), µ(-x)} = µ(x) for all x in M.

Now (ii) follows from (i).

The next part of this section deals with the concept “level submodule”.


2.7 Theorem: A fuzzy subset µ of a module M is a fuzzy submodule

⇔ the level set µt is a submodule of M for all t ∈ [0, µ(0)].

Proof: Proof is a straightforward verification.

2.8 Definition: Let µ be any fuzzy submodule of M. The submodules

µt, t ∈ [0, 1] where µt = {x ∈ M / µ(x) ≥ t} are called level submodules

of µ.

2.9 Result : Let M1 ⊆ M. Define µ(x) = 1 if x ∈ M1; µ (x) = 0

otherwise. Then the following conditions are equivalent:

(i) µ is a fuzzy submodule; and

(ii) M1 is a submodule of M.

Proof: (i) ⇒ (ii): Let x, y ∈ M1. Now µ(x) = µ(y) = 1.

Now µ(x - y) ≥ min{µ(x), µ(y)} (since µ is fuzzy submodule)

= min{1, 1} = 1 ⇒ µ(x - y) ≥ 1 ⇒ x - y ∈ M1.

Let r ∈ R, x ∈ M1. Now µ(rx) ≥ µ(x) = 1 (since µ is fuzzy submodule)

⇒ µ(rx) ≥ 1 ⇒ rx ∈ M1. Therefore M1 is a submodule of M.

(ii) ⇒ (i): Let x, y ∈ M. If x, y ∈ M1, then x - y ∈ M1 and so µ(x - y)

= 1 ≥ min{1, 1} = min{µ(x), µ(y)}. If x ∈ M1 and y ∉ M1, then

x - y ∉ M1 and so µ(x - y) = 0 ≥ min{1, 0} = min{µ(x), µ(y)}.

If x ∉ M1, y ∉ M1, then µ(x - y) ≥ 0 = min{µ(x), µ(y)}. Let r ∈ R.

If x ∈ M1, then rx ∈ M1, and so µ(rx) = 1 = µ(x). If x ∉ M1, then

µ(rx) ≥ 0 = µ(x). Hence µ is a fuzzy submodule.


2.10 Proposition: Let µ be a fuzzy submodule of M and µt, µs (with

t < s) be two level submodules of µ. Then the following two conditions

are equivalent:

(i) µt = µs; and (ii) there is no x ∈ M such that t ≤ µ(x) < s.

Proof: (i) ⇒ (ii): In a contrary way, suppose that there exists an element

x ∈ M such that t ≤ µ(x) < s. Then x ∈ µt and x ∉ µs and so µt ≠ µs,

a contradiction. Hence we get (ii).

(ii) ⇒ (i): Since t < s we have µt ≥ µs. Let x ∈ µt ⇒ µ(x) ≥ t. By

given condition (ii), there is no y such that s > µ(y) ≥ t and so µ(x) ≥ s

which implies x ∈ µs. Thus µt ≤ µs.

3. Minimal Elements

We start this section by introducing the new concept "minimal

element".

3.1 Definition: An element x ∈ M is said to be a minimal element if

the submodule generated by x is minimal in the set of all non-zero

submodules of M.

3.2 Theorem: If M has DCC on its submodules, then every nonzero

submodule of M contains a minimal element.


Proof: Let K be a nonzero submodule of M. Since M has DCC on its

submodules, it follows that K has DCC on its submodules. So K

contains a minimal submodule A (that is, A is minimal in the set of all

non-zero submodules of K). Let 0 ≠ a ∈ A. Then 0 ≠ Ra ⊆ A and so

Ra = A. Since Ra is a minimal submodule, we have that a is a

minimal element.

3.3 Note: There are modules which do not satisfy DCC on its

submodules, but contains a minimal element. For this we observe the

following example.

3.4 Example: Write M = Ζ ⊕ Ζ6. Now M is a module over the ring

R = Ζ. Clearly M have no DCC on its submodules. Consider a = (0, 2)

∈ M. Now the submodule generated by a, that is, Ζa = {(0, 0), (0, 2),

(0, 4)} is a minimal element in the set of all non-zero submodules of M.

Hence a is a minimal element.

3.5 Theorem: Every minimal element is an u-element.

Proof: Let 0 ≠ a ∈ M be a minimal element. Consider Ra. Let 0 ≠ L

and B be submodules of M such that L ⊆ Ra, B ⊆ Ra and

L ∩ B = (0). Since L ≠ 0, 0 ⊆ L ⊆ Ra, and a is minimal, it follows that

L = Ra. Now B = B ∩ Ra = B ∩ L = (0). This shows that L is

essential in Ra. Hence Ra is uniform submodule and so a is an

u-element.


3.6 Note: (i) The converse of Theorem 3.5 is not true. For this observe

the example 3.7 (given below).

(ii) If M is a vector space over a field R, then every non-zero element

is a minimal element as well as an u-element.

3.7 Example: Write M = Ζ as a module over the ring R = Ζ. Since Ζ

is a uniform module, and 1 is a generator, we have that 1 is an u-element.

But 2Ζ is a proper submodule of 1.Ζ = Ζ = M. Hence 1 cannot be a

minimal element. Thus 1 is an u-element but not a minimal element.

3.8 Theorem: Suppose µ is a fuzzy submodule of M.

(i) If a ∈ M, then for any x ∈ Ra we have µ(x) ≥ µ(a); and

(ii) If a is a minimal element, then for any 0 ≠ x ∈ Ra we have

µ(x) = µ(a).

Proof: (i) is clear from the definition of fuzzy submodule.

(ii) Let a ∈ M be a minimal element. Let 0 ≠ x ∈ Ra. Now

0 ≠ Rx ⊆ Ra and so Rx = Ra. Now a ∈ Rx and by (i), we have

µ(a) ≥ µ(x). Thus µ(x) = µ(a).

3.9 Lemma: If x is an u-element of a module M with DCC on

submodules, then there exist minimal element y ∈ Rx such that

Ry ≤e Rx.


Proof: Consider the submodule Rx. By Theorem 3.2, there exists a

minimal element y ∈ Rx. Since Ry is a non-zero submodule of Rx, and

Rx is uniform submodule, it follows that Ry ≤e Rx.

3.10 Theorem: If M has DCC on its submodules, then there exist

linearly independent minimal elements x1, x2, ….., xn in M where

n = dim M, and the sum <x1> + …….+ <xn> is direct and essential in M.

Also B = {x1, x2,….,xn} forms a basis for M.

Proof: Since M has DCC on its submodules, it follows that M has

FGD. Suppose n = dim M. Then by Result 1.5 of Satyanarayana [ 14 ],

there exist u-linearly independent elements u1, u2, …., un such that the

sum <u1> +…… + <un> is direct and essential in M. Since M has DCC,

by Lemma 3.9, there exist minimal elements xi ∈ Rui (Since R contains

1) such that Rxi ≤e Rui for 1 ≤ i ≤ n. Since u1, u2, ….., un are linearly

independent, it follows that x1, x2, ….., xn are also linearly independent.

Thus we got linearly independent minimal elements x1, x2, ….., xn in M

where n = dim M. Since Rxi ≤e Rui by a known result mentioned in the

introduction, it follows that Rx1⊕…..⊕Rxn ≤e Ru1⊕…… Ru⊕ n ≤e M

and so Rx1⊕…..⊕Rxn ≤e M. Thus B = { x1, x2,….,xn }forms a basis

for M.


4. Fuzzy Linearly Independent Elements

4.1 Definition: Let M be a module and µ be a fuzzy submodule of M.

x1, x2, …, xn ∈ M are said to be fuzzy µ-linearly independent ( or fuzzy

linearly independent with respect to µ) if it satisfies the following two

conditions: (i) x1, x2, …, xn are linearly independent; and

(ii) µ(y1 + … + yn) = min{µ(y1), …, µ(yn)} for any yi ∈ Rxi, 1 ≤ i ≤ n.

4.2 Theorem: Let µ be a fuzzy submodule on M. If x1, x2, …, xn are

minimal elements in M with distinct µ-values, then x1, x2, …, xn are

(i). linearly independent; and (ii). fuzzy µ-linearly independent.

Proof: The proof is by induction on n. If n = 1, then x1 is linearly

independent and also fuzzy linearly independent. Suppose the statement

is true for (n - 1). Now suppose x1, x2, …, xn are minimal elements

with distinct µ values. By induction hypothesis x1, x2, …, xn-1 are

linearly independent and fuzzy linearly independent. If x1, …, xn are not

linearly independent, then the sum of Rx1, Rx2, …, Rxn-1, Rxn is not

direct. This means Rxn ∩ (Rx1 + … + Rxn-1) ≠ 0 ⇒ 0 ≠ rnxn =

r1x1 + … + rn-1xn-1 for some ri ∈ R, 1 ≤ i ≤ n. µ(xn) = µ(rnxn) (use the

fact xn is a minimal element and Theorem 3.8) = µ(r1x1 + … + rn-1xn-1)

= min{µ(r1x1), …, µ(rn-1xn-1)}(by induction hypothesis) = µ(rixi) for

some fixed i (1 ≤ i ≤ n). Now 0 ≠ µ(rixi) = µ(xi) (use the fact xi is

minimal, and use Theorem 3.8). Thus µ(xn) = µ(xi), a contradiction.

Hence x1, …, xn are linearly independent. Now we prove that x1,


x2,….,xn are fuzzy linearly independent. Suppose yi ∈ Rxi, 1 ≤ i ≤ n

and yi = sixi, with si ∈ R, 1 ≤ i ≤ n. µ(y1 + y2 + … + yn-1) = min{µ(y1),

…, µ(yn-1)} (by induction hypothesis) = µ(yj) (for some j with 1 ≤ j ≤ n

-1) = µ (sjxj) = µ(xj) (by Theorem 3.8).

Now µ(xj) ≠ µ(xn) ⇒ µ(y1 + … + yn-1) = µ (xj) ≠ µ (xn) = µ (yn).

⇒ µ(y1 + … + yn-1 + yn) = min{µ(y1 + … + yn-1), µ(yn)} (by Proposition

2.4) = min{min{µ(y1), …, µ(yn - 1)}, µ(yn)} = min{µ(y1), ….…, µ(yn - 1),

µ(yn)}. This shows that x1, …, xn are fuzzy linearly independent with

respect to µ.

5. Fuzzy Dimension

We start this section by defining the concept “fuzzy pseudo

basis”.

5.1 Definition: (i). Let µ be a fuzzy submodule of M. A subset B of M

is said to be a fuzzy pseudo basis for µ if B is a maximal subset of M

such that x1, x2,….., xk are fuzzy linearly independent for any finite

subset { x1, x2,….., xk } of B.

(ii). Consider the set Β = {k / there exist a fuzzy pseudo basis B for µ

with |B| = k}. If Β has no upper bound then we say that the fuzzy

dimension of µ is infinite. We denote this fact by S-dim(µ) = ∞. If Β

has an upper bound, then the fuzzy dimension of µ is sup Β. We denote

this fact by S-dim(µ) = sup Β. If m = S-dim(µ) = sup Β, then a fuzzy


pseudo basis B for µ with |B| = m, is called as fuzzy basis for the fuzzy

submodule µ.

5.2 Result: Suppose M has FGD and µ is a fuzzy submodule of M.

Then (i). |B| ≤ dim M for any fuzzy pseudo basis B for µ; and

(ii). S-dim (µ) ≤ dim M.

Proof : Suppose n = dim M.

(i). Suppose B is a fuzzy pseudo basis for µ. If | B | > n, then B contain

distinct elements x1, x2, ….., xn + 1. Since B is a fuzzy pseudo basis, the

elements x1, x2, ….., xn + 1 are linearly independent; and by Result 1.1

of Satyanarayana [ 14 ], it follows that n + 1 ≤ n, a contradiction.

Therefore | B | ≤ n = dim M.

(ii). From (i) it is clear that dim M is an upper bound for the set

Β = {k / there exist a fuzzy pseudo basis B for µ with |B| = k}.

Therefore S-dim(µ) = sup Β ≤ dim M.

5.3 Definition: A module M is said to have a fuzzy basis if there exists

an essential submodule A of M and a fuzzy submodule µ of A such that

S-dim(µ) = dim M. The fuzzy pseudo basis of µ is called as fuzzy basis

for M.

5.4 Remark: If M has FGD, then every fuzzy basis for M is a basis for

M.


5.5 Theorem: Let M be a module with DCC on submodules. Then M

has a fuzzy basis (In other words, there exists an essential submodule A

of M and a fuzzy submodule µ of A such that S-dim(µ) = dim M).

Proof: Since M has DCC on submodules, it has FGD. Suppose dim M

= n. By Theorem 3.10, there exists linearly independent minimal

elements x1, x2,……xn such that { x1, x2,……xn }forms a basis for M.

Take 0 ≤ t1 < t2 .….< tn ≤ 1. Define µ (yi) = ti for yi ∈ Rxi. Then µ is

fuzzy submodule on A = Rx1 + Rx2 + ……+ Rxn ≤e M. By

Theorem 4.2, x1, x2,……xn are fuzzy µ- linearly independent. So {x1,

x2,……xn} is a pseudo basis for µ. Now dim M = n ≤ sup Β ≤ dim M

(by Result 5.2) and hence S-dim(µ) = dim M. This shows that M has a

fuzzy basis.

Acknowledgements

Part of this paper done by first two authors at Universitat Bundeshwer, Hamburg while they are attending the 18th International Conference on Near-rings & Near-fields, Hamburg, July 27-Aug03, 2003. The first author is thankful to University Grants Commission, New Delhi for the financial support under the grant No. F.8-8/2004(SR). The first author also acknowledges his thanks to Prof. Dr. L. Venugopal Reddy, the Vice-Chancellor; Prof. Dr. V. Luther Das, Registrar; Prof. Dr. P. Thrimurthy, Department of Computer Science, of Acharya Nagarjuna University; and Prof. Dr. P. V. Arunachalam (Former Vice-Chancellor, Dravidian University) for their constant encouragement.


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ON FUZZY DIMENSION OF A MODULE WITH DCC ON SUBMODULES

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