Cancellation On Fuzzy Projective Modules And Schanuel’s

Lemma Using Its Conditioned Class

AMARJIT KAUR SAHNI1, JAYANTI TRIPATHI PANDEY2,

RATNESH KUMAR MISHRA3

1,2Department of Mathematics, AIAS, Amity University, Uttar Pradesh, INDIA

3Department of Mathematics, NIT, Jamshedpur, INDIA

Abstract:–As an extension, the current study looks at fuzzy projective module cancellation and fuzzy module

equivalence in speciﬁc situations. While addressing cancellation, we provide the necessary and suﬃcient

criteria for fuzzy projective modules to fulﬁll cancellation over the polynomial ring and ring R. Furthermore,

using fuzzy p-poor modules, we have established an intriguing result in Schanuel’s lemma, claiming that

for any two fuzzy exact sequences of fuzzy R-modules 0 →µ1

−→ η1

¯g1

−→ µ→0 and 0 →µ2

−→ η2

¯g2

−→ µ→0.

If η1and η2are fuzzy p-poor modules then µ1⊕η2∼

=µ2⊕η1. The same is reinforced by an acceptable

illustration of fuzzy p-poor module.

Key-words:- fuzzy modules, fuzzy projective module, fuzzy projective poor-module, fuzzy subprojective

poor-module, schanuel’s lemma.

Received: August 27, 2021. Revised: May 19, 2022. Accepted: June 9, 2022. Published: July 1, 2022.

1 Introduction

Throughout the study, rings are commutative with

identity, and modules are unitary. Authors like

Gilmer[6] studied the ring whose ideals meet can-

cellation characteristics independently. According

to his research, every ring ideal is conﬁned cancel-

lation if and only if the ring is a nearly Dedekind

domain or a primary ring. D.D. and D.F. Ander-

son[2] veriﬁed a similar ﬁnding and investigated

it further. Mijbass[16] generalized this notion to

modules. Many researchers worked on its various

types, as mentioned in [5], [8], [25]. Also, weak

cancellation modules by Naoum and Mijbas[18]

proved some properties of them as well as their

relations with other types of modules, such as

projective and ﬂat modules, and provided some

conditions under which projective and ﬂat mod-

ules act as weak cancellation modules. Zhang and

Tong[24] also worked on the characterization of

the cancellation property for projective modules

and demonstrated that Dedekind domains contain

it. Bothaynah, Khalaf and Mahmood investigated

purely and weakly purely cancellation modules

in [3] and developed equivalent criteria for each

kind. Mahmood, Bothaynah and Rasheed[15] in-

vestigated comparable cancellation modules and

discovered some connections between them and

cancellation modules. They also looked at the im-

pact of module localisation and tracing on this sort

of module. On the Laurent polynomial ring, au-

thors like Mishra[17] studied cancellation modules.

Later, as illustrated in [11], [12] and [4] cancel-

lation modules such as purely, restricted, weakly

restricted, fully and naturally were fuzziﬁed.

Since then, the current work has focused on ei-

ther the classical version of cancellation on projec-

tive modules or various sorts of cancellation fuzzy

modules. Thus, the current work addresses the

gap, and we extend the existing situation by exam-

ining cancellation on fuzzy projective modules and

demonstrating the equivalence of fuzzy modules us-

ing Schanuel’s lemma. To demonstrate Schanuel’s

lemma exemption for fuzzy projective modules, we

constructed a new structure called the fuzzy p-poor

module. The current research is organised as fol-

lows. In Section 2, the basic deﬁnitions are given

for a better understanding of the reader. Section

3 is motivated by [17] and deals with the cancel-

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lation of fuzzy projective modules over polynomial

rings. In it, while extending the interesting results

to their fuzzy framework we have discussed the

fuzzy version of Schanuel’s Lemma which shows the

equivalence of two fuzzy modules µ1and µ2pro-

vided that there exist two fuzzy projective modules

η1and η2such that µ1⊕η2∼

=µ2⊕η1⇒µ1∼

=µ2.

This lemma shows how far modules are being pro-

jective and also is useful in deﬁning the Hellar op-

erator in the stable category and giving an elemen-

tary description of the dimension shifting. In ad-

dition to the above necessary and suﬃcient con-

ditions for which a fuzzy projective module has a

cancellation property are discussed in the section.

Finally, section 4 draws the attention of the reader

to the introduction of a fuzzy structure called the

fuzzy p-poor module, which exempts the require-

ment of fuzzy projective modules in the Schanuel’s

lemma and also discusses the few relevant and in-

teresting properties of the same.

2 Preliminaries

The following sections outline the deﬁnitions and

outcomes used in this research.

This paper’s Terminology is as follows:

1. RMand MRdenote the left and right R –

module respectively for each module M.

2. ∃means there exists

3. µMdenotes the fuzzy module µover module

4. ⇒means implies

5. µ(m) represents the arbitrary element of

fuzzy set µM.

6. µtdenotes the level subset of a fuzzy module

µ.

Deﬁnition 2.1.[13] If the following conditions are

met, a fuzzy subset µMis called a fuzzy submodule

of module M:

(i) µ(m+n)≥min{µ(m), µ(n)}

(ii) µ(xm)≥µ(m), for all m, n ∈Mand x∈R

(iii) µ(−x) = µ(x) for all x∈M

(iv) µ(0) = 1

Deﬁnition 2.2 [14] A fuzzy R-module µPis called

projective if and only if for every surjective fuzzy

R-homomorphism ¯

f:µA→µBand for every fuzzy

R-homomorphism ¯g:µP→µBthere exists a fuzzy

R-homomorphism ¯

h:µP→µAsuch that the ﬁgure

below commutes that is : ¯

f¯

h=¯g

Fig.1 Fuzzy Projective Module

Note : We can also decipher the above deﬁnition

as µPis µA- projective.

Deﬁnition 2.3[Classical Version][5] A module

M is said to have a cancellation property if for all

modules H and K, A ⊕H∼

=A⊕K implies H ∼

Lemma 2.4[Classical Version of Schanuel’s

Lemma in Projective Modules] [10] Let R be

a ring. Then for any given exact sequences of R-

modules 0→M1

−→ P1

−→ M→0and 0→M2

−→

−→ M→0with P1and P2projective we have

M1⊕P2∼

=M2⊕P1.

Deﬁnition 2.5[19] A module M is deﬁned to be

projectively poor(or p-poor) if its domain of pro-

jectivity contains only semisimple modules. Where

N(M) = [N ∈MR|M is N-projective] is deﬁned

as a projectivity domain of module M.

Deﬁnition 2.6[23] The sequence .... →

µn−1

fn−1

−−−→µn

−→µn+1 →.... of R- fuzzy module ho-

momorphism is termed as fuzzy exact if and only

if Im ¯

fn−1= Ker ¯

fnfor every n. Here Im ¯

fn−1and

Ker ¯

fnmeans µn|Imfn−1and µn|Kerfnthat is

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µnis restricted to image and kernel respectively.

Deﬁnition 2.7[23] The exact sequence of the form

0→µA

−→ ηB

−→ νB→0is called as fuzzy short

exact sequence.

Deﬁnition 2.8[25] A ﬁnitely generated projective

R-module P is said to be cancellative if P ⊕Rn∼

Q⊕Rnimplies P ∼

Deﬁnition 2.9[25] Let R be a ring and P be a

projective R-module. An element p ∈P is called

unimodular if there is a surjective R-linear map

ϕ:P→Rsuch that ϕ(p) = 1.

Proposition 2.10[8] Let µbe a fuzzy module of an

R- module M,then µis a fuzzy cancellation module

if and only if µtis a cancellation module.

3 Cancellation on Polynomial Rings

Let R be a ring and µMbe the fuzzy R-module. Let

µM[X] be the fuzzy polynomial module over poly-

nomial ring R[X] where µM[X] = [ΣaiXi, where,

aiare fuzzy numbers]. If ¯g:µM→ηNis a

fuzzy homomorphism of fuzzy R-modules then it

induces a homomorphism Ψ : µM[X]→ηN[X]

deﬁned as Ψ(ΣaiXi) = Σ¯g(ai)Xi. Given any fuzzy

R-module µand ¯

f∈End(µ). We can make µas

R[X] module whose scalar multiplication is deﬁned

as (mΣaiXi) = aiΣ¯

fn(m) and denote the R[X]

module as ¯

fµ. Then there is canonical R[X] sur-

jection ϕ¯

f:µ[X]→¯

fµdeﬁned as ϕ¯

f(ΣaiXi) =

Σ¯

fn(m).

We have extended Schanuel’s lemma (described

in 2.4) to its fuzzy environment, having fuzzy

projective-modules, in the following lemma. It

is linked to the equivalence of two fuzzy modules

µM1and µM2if two fuzzy projective modules µP1

and µP2are present such that µM1⊕µP2∼

=µM2⊕

µP1.

Lemma 3.1[Fuzzier form of Schanuel’s

Lemma]Given the two sequences of fuzzy R-

modules

0→µ1

−→ η1

¯g1

−→ µ→0and 0→µ2

−→ η2

¯g2

−→

µ→0. If they are fuzzy exact with η1and η2are

fuzzy projective modules then µ1⊕η2∼

=µ2⊕η1.

Proof. Fuzzy direct sum η1⊕η2can be formed us-

ing fuzzy R-modules η1and η2. Next ν=

η1⊕η2= [(η1(x1), η2(x2)) ∈η1⊕η2: ¯g1(η1(x1))

= ¯g2(η2(x2))]. Clearly, ν⊆η1⊕η2and νis non-

empty set. Then for each (η1(x1), η2(x2)) and

(η1(y1), η2(y2)) in νand r in R we have

¯g1[(η1(x1)) + (η1(y1))] = ¯g1(η1(x1)) + ¯g1(η1(y1))

= ¯g2(η2(x2)) + ¯g2(η2(y2))

= ¯g2[(η2(x2)) + (η2(y2))]

⇒[(η1(x1)) + (η1(y1))] ∈ν.

and [(η1(x1))r+ (η2(x2))r]∈νor in other words

we can say that νis a submodule η1⊕η2. Next we

have ¯g1is surjective homomorphism so ¯g1(η1) = µ

therefore for each ¯g1(η1)∈µ∃η2(x)∈η2such that

¯g1(η1(x)) = ¯g2(η2(x)). Deﬁned homomorphism ¯π1

:ν→η1deﬁned as ¯π1(η1, η2) = η1. Then we have

ker¯π1= [(η1, η2) : ¯π1(η1, η2) = 0]

= [(η1, η2) : η1= 0]

= [(0, η2) : ¯

g1(η2)=0

=ker¯g2

=Im ¯

f2.

Im ¯

f2=µ2because ¯

f2is an injective homomor-

phism. As a result, we have Ker ¯π1=µ2is ob-

tained. Thus, a fuzzy short exact sequence can be

created 0 →µ2→ν¯π1

−→ η1→0————-(1)

Since η1is fuzzy projective equation (1) splits thus

∃¯

h:η1→νsuch that ¯π1o¯

h=Idη1. Hence

by [22] ν=η1⊕µ2. A fuzzy short exact sequence

0→µ1→ν¯π2

−→ η2→0——–(2) can be generated

in a similar way to give ν=η2⊕µ1. Therefore

µ1⊕η2∼

=µ2⊕η1.

Proposition 3.2 Let µand µ′be the fuzzy pro-

jective R[X] modules and ¯

ϕ:µ′→µ,¯

ψ:µ→µ′

be the fuzzy injective homomorphism. If R[X],

µ/¯

ϕ¯

ψµ,µ′/¯

ψ¯

ϕµ′are fuzzy projective over R then

µ/¯

ϕµ′and µ′/¯

ψµ are also fuzzy projective over R.

Proof. : Since µand µ′are fuzzy R[X] projective

and R[X] is R projective, there exists fuzzy R pro-

jective module ηand fuzzy R[X] projective mod-

ules µ1and µ′

1and. We now, get for some positive

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integers n and m

µ⊕µ1∼

=R[X]n

=µ′⊕µ′

1∼

=R[X]m

=R[X]⊕η∼

=XR

=R[X]n⊕ηn∼

=XR

=µ⊕µ1⊕ηn∼

=XR

=µ⊕ηn

1∼

=XRwhereµ1⊕ηn

=ηn

Therefore µis fuzzy R projective. Similarly µ′

is fuzzy R projective. Since 0 →µ¯

−→ µ′→

µ′/¯

ψµ →0, pd(µ′/¯

ψµ)≤1. Similarly, pd(µ/ ¯

ϕµ′)

≤1. Now isomorphism between µand ¯

ψµ induces

µ/ ¯

ϕµ′∼

=¯

ψµ/ ¯

ψ¯

ϕµ′we have exact sequences

Fig.2 Fuzzy Exact sequences

By Lemma 3.1 µ′/¯

ψ¯

ϕµ′⊕µ∼

=¯

ψµ/ ¯

ψ¯

ϕµ′⊕µ′so that

µ′/¯

ψ¯

ϕµ′⊕µ∼

=µ/ ¯

ϕµ′⊕µ′. Direct sum of fuzzy

projectives are also fuzzy projective and µ′/¯

ψ¯

ϕµ′,

µare fuzzy R-projective, µ′/¯

ψ¯

ϕµ′⊕µare fuzzy

R-projective. Therefore µ/ ¯

ϕµ′⊕µ′are fuzzy R-

projective. Thus, µ/ ¯

ϕµ′⊕µ′⊕µ0∼

=Rncomes

from the deﬁnition of fuzzy R-projective module

where µ0is a fuzzy R-module. Let ¯µ=µ′⊕µ0be

a fuzzy R-module. Then µ/ ¯

ϕµ′⊕¯µ∼

=Rn. Hence

µ/ ¯

ϕµ′is a fuzzy R-projective. Similarly, µ′/¯

ψµ is

also fuzzy R-projective.

Corollary 3.3 Let µand µ′be fuzzy projective

R[X] modules with µ⊃µ′⊃fµ, where f is a monic

polynomial of polynomial ring. If R[X] and R[x]/f

R[X] are R-projective then µ/µ′is fuzzy projective.

Proof. Let us assume the inclusion map ¯

ϕ:µ′→

µand ¯

ψthe multiplication by f from µ→µ′. Then

µ/ ¯

ϕ¯

ψµ =µ/fµand µ′/¯

ψ¯

ϕµ′=µ′/fµ′. Since µand

µ′are fuzzy projective R[X] projective and R[X],

R[X]/fR[X] are R-projective, thus µ/fµand µ′/fµ′

are R[X]/fR[X] are projective. Hence µ/fµand

µ′/fµ′are fuzzy R-projective. Thus, µ/ ¯

ϕ¯

ψµ and

µ′/¯

ψ¯

ϕµ′are also fuzzy R-projective. By preposi-

tion 3.2 fuzzy module µ/µ′is fuzzy projective.

Proposition 3.4 Let µbe a fuzzy R-module

and ¯

f∈End(µ). Then 0→µ[X]X.1µ[X]−f[X]

−−−−−−−−→

µ[X]¯

ϕf

−→ ¯

fµ→0is an fuzzy exact sequence of R[X]

modules.

Proof. Clearly ¯

ϕfis surjective so we have

ϕf(X.1µ[X]−f[X])(XAiXi)

=¯

ϕf(X(AiXi+1 −f(Ai)Xi

=¯

ϕf[X(AiXi+1 −Xf(Ai)Xi]

=X(fi+1(Ai) + fi(f(Ai)))

=X(fi+1(Ai) + fi+1(Ai))

= 0.

Thus, Im(X.1µ[X]−f[X]) ⊆Ker ¯

ϕf. Now, to

show Ker ¯

ϕf⊆Im(X.1µ[X]−f[X]). Let PAiXi

∈Ker ¯

ϕf⇒¯

ϕf(PAiXi) = 0. Then,

Z=Z−Pfi(Ai)

=P(AiXi−fi(Ai)

=P(Xi.1µ[X]−fi)Ai

= (Xi.1µ[X]−f[X])[..1/(X−f)(Xi−fi)/Xifi−

1/(X−f)(Xi−1−fi−1)/Xi−1fi−1.. −..1/(X−

f)(X−f)/(X−f)+0+(X−f)/(X−f) + ..]Ai

= (Xi.1µ[X]−f[X]) Phi(Ai)

Z∈Im(X.1µ[X]−f[X]).

Hence, Ker ¯

ϕf⊆Im(X.1µ[X]−f[X]). Thus,

Ker ¯

ϕf= Im(X.1µ[X]−f[X]).

Theorem 3.5 Let µand µ′be ﬁnitely generated

fuzzy projective R[X] modules. Suppose µ⊃µ′⊃

fµ for some monic polynomial f∈R[X]. Then

µand µ′are stably isomorphic. In Particular, if

µf∼

=µ′

f′then µand µ′are stably isomorphic.

Proof. : Take η=µ/µ′. R[X]/fis a free R-module

since fis a monic polynomial. As a result, µ/fµ

is R-projective, with corollary 3.3 indicating that

ηis fuzzy R-projective. We have a fuzzy exact se-

quence of R[X] modules

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Fig.3 Fuzzy Exact Sequence

Since µ′⊂µand η=µ/µ′. The ﬁrst sequence is

fuzzy exact since ¯

iand ¯πare inclusive and surjec-

tive maps respectively. By preposition 3.4 the sec-

ond sequence is also fuzzy exact. Since ηis fuzzy

projective, η[X] is fuzzy R[X] projective. Thus by

Schanuel’s Lemma µ⊕η[X]∼

=µ′⊕η[X]. Hence µ

and µ′are stably isomorphic.

3.1 Cancellation on Ring R

All rings considered in this section are associative

with identity and modules are unital right mod-

ules.

Example 3.1.1. Let µ:Z→[0,1] is deﬁned as

µ(z) = (1,if z∈2Z

0,elsewhere

Then µis a fuzzy module for all z ∈Z. Further-

more, because µt= 2Z is a cancellation module,

µcan be regarded as a cancellation fuzzy mod-

ule[proposition 2.10].

We’ve now deﬁned the necessary and suﬃcient re-

quirements for fuzzy projective modules to have

the cancellation property.

Proposition 3.1.2 Let R be a ring, µbe a fuzzy

projective R-module and ¯

ϕ= End(µ). If η∼

=⊕R/ν

for some index set N and νfuzzy submodule of ⊕

R then the following are indistinguishable:

1. For any fuzzy R module ψ

µ⊕η∼

=µ⊕ψ⇒η∼

=ψ.

2. Whenever ¯

θ¯

λ+ ¯α¯σ= 1µ∈¯

ϕwith ¯α(¯ν) = 0.

Where ¯

θ,¯

λ∈¯

ϕ, ¯α= (µ1, µ2, .....µi, ....)∈

Qµi∼

=Hom(⊕R, µ) and ¯σ∈Hom(µ, ⊕R),

there are ¯τ1∈Qµand ¯

h∈Hom(⊕R, ⊕R)

with ¯τ1(ν) = 0 and ¯

h(ν)⊆νsatisfying the

following conditions:

(i) ¯

θ¯τ1+ ¯α¯

h= 0.

(ii) If ¯

θ(µ1) + ¯α(r) = 0 where (µ1∈µand

r∈ ⊕Rthen there is z ∈ ⊕Rsuch that

µ1∈¯τ1(z) and r- ¯

h(z)∈ν.

(iii) ¯τ1(r) = 0 and ¯

h(r)∈ν⇒r∈νfor any

r∈ ⊕R.

Proof. : Write F = ⊕R. Since η=F/ν, there is

surjective homomorphism ¯q:F→ηwith Ker¯q=

ν. So we have the following fuzzy exact sequence.

Since 0 →ν→F¯q

−→ η→0. (i) ⇒(ii) Let us assume

θ¯

λ+ ¯α¯σ= 1µ∈¯

ϕwith ¯α(¯ν) = 0 then there is ¯pin

Hom (η, µ) such that ¯α= ¯p¯q, so ¯

θ¯

λ+ ¯p¯q¯σ=1. Set

¯π= (¯

θ, ¯p)∈Hom (µ⊕η, µ) and ¯π= (¯

λ, ¯q¯σ)∈Hom

(µ, µ ⊕η) then ¯π¯

ϕ= 1µwhich mean the following

fuzzy exact sequence splits.0 →kerπ→µ⊕η¯π

−→

µ→0. Hence µ⊕η∼

=µ⊕ker¯π. By (1) we have

η∼

=ker¯πwhich implies there is a homomorphism

¯τ∈Hom (η, µ ⊕η) such that following sequence is

fuzzy exact. 0 →η¯τ

−→µ⊕η¯π

−→µ→0—–(1). Write

¯τ= (¯τ′

1,¯τ′

2) for some ¯τ′

1∈Hom (η, µ) and ¯τ′

2∈

Hom (η, η). Take ¯τ1= ¯τ′

1¯q∈Hom (F, µ) and ¯τ2

= ¯τ′

2¯q∈Hom (F, µ). Then ¯τ1ν= 0. Since F is

projective, there is ¯

h∈Hom(F, F) such that ¯τ2

= ¯τ′

2¯q= ¯q¯

h. Thus, ¯q¯

h(ν) = ¯τ′

2¯q(ν)=0so¯

h(ν)⊆

Ker ¯q=ν. Since the sequence (1) is fuzzy exact,

we have ¯π¯τ= 0 that is ¯

θ¯τ′

1+ ¯p¯τ′

2= 0. So ¯

θ¯τ′

¯α¯

h=¯

θ¯τ′

1+ ¯p¯q¯

h=¯

θ¯τ′

1¯q+ ¯p¯τ′

2¯q= 0. Thus (1) holds

Now, ker¯π⊆¯τ(η). Suppose µ1(x)∈µand r ∈

F with ¯

θ(µ1) + ¯p(¯q(r)) = 0, that is (µ1(x) + ¯q(r)∈

Ker¯π= ¯τ(η). So there is ¯q(z)∈ηwith z ∈F. Such

that µ1(x) = ¯τ′

1(¯q(z)) = ¯τ1(z) and

¯q(r) = ¯τ′

2(¯q(z))

= ¯q¯

h(z)

=r−¯

h(z)∈ker¯q

=ν

Thus(ii) holds. Moreover, ¯τis a monomorphism.

For any r ∈F with ¯τ1(r) = 0 and ¯

h(r)∈ν, we have

¯τ1¯q(r) = (¯τ1(r),¯q¯

h(r)) = 0. So, ¯q(r) = 0. Thus,

(iii) holds.

(2) ⇒(1) Suppose µ⊕η∼

=µ⊕ψ. Then there is a

fuzzy split exact sequence 0 →ψ→µ⊕η¯π

−→µ→0

with ¯π¯

ξ= 1µfor some ξ∈Hom(µ, µ ⊕η). Set ¯π

= (¯

θ, ¯p) and ¯

ξ= (¯

λ, ¯

ϕ1) for some ¯

θ, ¯

λ∈¯

ϕ. ¯p∈

Hom (η, µ) and ¯

ϕ1∈Hom (µ, η). Since we have

µis fuzzy projective ∃¯σ∈Hom (µ, F) such that

ϕ1= ¯q¯σ. Take ¯α= ¯p¯q∈Hom (µ, F) ∼

=Qµi, i ∈N

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then ¯α(ν) = ¯p¯q(ν) = 0 and

θ¯

λ+ ¯α¯σ=¯

θ¯

λ+ ¯p(¯q¯σ)

=¯

θ¯

λ+ ¯p¯

ϕ1

= ¯π¯

= 1µ

By hypothesis there is ¯τ1∈Hom(F, µ) and ¯

h∈

Hom(F, F) with ¯τ1(ν) = 0 and ¯

h(ν) is a fuzzy sub-

module of νsatisfying the condition(i)-(iii). Since

¯τ1(ν) = 0 we have ¯τ′

1∈Hom (η, µ) such that

¯τ1= ¯τ′

1¯q. Now,

h(ν)⊂ν

⇒¯q¯

h(ν)=0

So there exists ¯τ′

2∈Hom(η, η) such that ¯q¯

h= ¯τ′

2¯q.

If ¯τ= (¯τ′

1,¯τ′

2)∈Hom(η, µ ⊕η) then conditions(i)

–(iii) gives the sequence 0 →η¯τ

−→µ⊕η¯π

−→µ→0 is

fuzzy exact. Therefore η∼

=Ker¯π∼

=ψ.

We can derive the following theorem from

the above preposition :

Theorem 3.1.3 Let R be a ring, µbe a fuzzy

projective R-module, and ¯

ϕ= End(µ) for some in-

dex set N then the following are indistinguishable:

1. For any fuzzy R module ψand any fuzzy

N-generated R module η,µ⊕η∼

=µ⊕

ψ⇒η∼

=ψ.

2. Whenever ¯

θ¯

λ+ ¯α¯σ= 1µ∈¯

ϕwith ¯α(¯ν) = 0.

Where ¯

θ,¯

λ∈¯

ϕ, ¯α= (µ1, µ2, .....µi, ....)∈

Qµi∼

=Hom(⊕R, µ) and ¯σ∈Hom(µ, ⊕R),

if νis fuzzy submodule of Ker ¯α, there is

¯τ1∈Qµand ¯

h∈Hom(⊕R, ⊕R) with ¯τ1(ν)

= 0 and ¯

h(ν)⊆νsatisfying the following

conditions:

(i) ¯

θ¯τ1+ ¯α¯

h= 0 ∈Hom(⊕R, µ).

(ii) If ¯

θ(µ1) + ¯α(r) = 0 where µ1∈µand

r∈ ⊕Rthen there is z ∈ ⊕Rsuch that

µ1∈¯τ1(z) and r - ¯

h(z)∈ν.

(iii) ¯τ1(r) = 0 and ¯

h(r)∈ν⇒r∈νfor any

r∈ ⊕R.

Note : A fuzzy projective R module µsatisfy the

cancellation property if and only if µsatisﬁes con-

dition (2) of Theorem 3.1.3 for any N.

4 Fuzzy P-Poor Modules

This section presents a few intriguing results in ad-

dition to the pertinent result where fuzzy p- poor

modules are shown as an alternative to fuzzy pro-

jective modules in equivalence of modules.

Here Mod-FR denotes the category of all fuzzy

right R-modules over the ring R and SS Mod-FR

stands for fuzzy semisimple right R-modules.

Deﬁnition 4.1 A fuzzy R-module µPis called

p-poor if and only if for every fuzzy semisimple

module µAsatisﬁes for each surjective fuzzy R-

homomorphism ¯

f:µA→µBand for every fuzzy

R-homomorphism ¯g:µP→µBthere exist a fuzzy

R-homomorphism ¯

h:µP→µAsuch that the ﬁgure

below commutes that is : ¯

f¯

h=¯g.

Fig.4 Fuzzy p-poor module

Example 4.2 Let µ,ηand ϕbe fuzzy modules that

are deﬁned over Q√2, Q√3and Q√3/Q respec-

tively as

µ(x+y√2) = 









1,if x, y = 0

4/5,if x =0, y =0

1/2,if y =0

η(x+y√3) = 









1,if x, y = 0

1/2,if x =0, y =0

1/3,if y =0

and ϕ[(x+y√3)+Q] = η(x+y√3) ∀x, y ϵQ.

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Fig.5 µis fuzzy projective

Then µis fuzzy projective, where the mappings

f,¯

hand ¯gare ¯

f[µ(x+y√2)] = η(x+y√3),

¯g[η(x+y√3)]=[ϕ(x+y√3)+Q] and ¯

h[µ(x+y√2)]

= [ϕ(x+y√3)+Q] respectively.

Example 4.3 Using the same module µas in ex-

ample 4.2 above, where M = Q√2= Q ⊕√2Q is

semi simple. Furthermore, deﬁne µ1over Q as,

µ1(x) = (1,if x = 0

4/5,if x =0

and µ2over √2Q as

µ2(x) = (1,if x = 0

4/5,if x =0

Then µ1and µ2are respectively fuzzy modules over

Q and √2Q. Furthermore, µ=µ1⊕µ2establishes

that µis a semi-simple R-module over M. This is

now known as the fuzzy p-poor module.

Deﬁnition 4.4 For a fuzzy module µP,P(µP) =

[µA|µPis µA-projective] is deﬁned as a projectiv-

ity domain of µP.

Note : Recalling the following deﬁnitions given in

[9]

(a)Deﬁnition 4.5 µMis said to be simple fuzzy

left module if it has no proper submodules.

(b)Deﬁnition 4.6 µMis said to be semi-simple

fuzzy left module if whenever for νN, a strictly

proper fuzzy submodule of µMthere exist a strictly

proper fuzzy submodule ηPof µMsuch that µM=

νN⊕ηP.

Note : A ring is said to be semi-simple if, every

left-module over it is semi-simple.

Deﬁnition 4.7 A ring R is called fuzzy semisimple

artinian if any of the following equivalent condi-

tions hold: (i) RMis semisimple

(ii) MRis semisimple

(iii) SS Mod-FR = Mod-FR

Remark :The fuzzy p-poor module is a special

case of the fuzzy projective module as the projectiv-

ity domain of it consists of only fuzzy semisimple

modules over ring R.

Lemma 4.8 Let µMbe a ﬁnitely generated fuzzy

R-module which has a projective direct summand

of rank f >d = dim of Y (where Y is the space

whose each element is the fuzzy maximal ideal of

R) and let νQbe a ﬁnitely generated fuzzy p-poor

module. Then if ηM′is another fuzzy R-module we

have νQ⊕µM∼

=νQ⊕ηM′⇒µM∼

=ηM′.

Proof. : Since µQ⊕θQ′∼

=Rnfor some n and θQ′.

We can reduce, by induction on n to the case R

=µQ. Using the given isomorphism to identify R

⊕µMwith R ⊕ηM′, we can write βR⊕µM=

αR⊕ηM′with βand αbeing unimodular. Since α

is unimodular then there exists a τ∈group of R

automorphism of (βR⊕µM) with τα =β. There-

fore, µM∼

=(βR⊕µM)/βR = τ(αR⊕ηM′)/τ(α

R) ∼

=(αR⊕ηM′)/αR∼

=ηM′.

Lemma 4.9[Schanuel’s Lemma using

fuzzy p-poor modules]Given the two sequences

of fuzzy R-modules 0→µ1

−→ η1

¯g1

−→ µ→0and

0→µ2

−→ η2

¯g2

−→ µ→0. If they are fuzzy exact

with η1and η2are fuzzy p-poor modules then µ1⊕

η2∼

=µ2⊕η1.

Proof. : A fuzzy direct sum η1⊕η2can be formed

using fuzzy p-poor modules η1and η2.

Next ν=η1⊕η2= [(η1(x1), η2(x2)) ∈η1⊕η2:

¯g1(η1(x1)) = ¯g2(η2(x2))]———-(1)

Clearly, ν⊆η1⊕η2and νis a non-empty set.

Then for each (η1(x1), η2(x2)) and (η1(y1), η2(y2))

in νand r in R we have

¯g1[(η1(x1)) + (η1(y1))] = ¯g1(η1(x1)) + ¯g1(η1(y1))

= ¯g2(η2(x2)) + ¯g2(η2(y2))[by(1)]

= ¯g2[(η2(x2)) + (η2(y2))]

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implying [(η1(x1))+(η1(y1))] ∈ν[by the deﬁnition

of ν] and [(η1(x1))r+ (η2(x2))r]∈νor in other

words we can say that νis a submodule η1⊕η2.

Since every fuzzy submodule of a fuzzy semisim-

ple module is fuzzy semisimple, then we can say

νis fuzzy semisimple. Next, we have ¯g1is sur-

jective homomorphism so ¯g1(η1) = µtherefore for

each ¯g1(η1)∈µ∃η2(x)∈η2such that ¯g1(η1(x)) =

¯g2(η2(x)). Deﬁne homomorphism ¯π1:ν→η1as

¯π1(η1, η2) = η1———-(2). Then we have

ker¯π1= [(η1, η2) : ¯π1(η1, η2) = 0]

= [(η1, η2) : η1= 0][by(2)]

= [(0, η2) : ¯

g2(η2) = 0][sinceη1= 0]

=ker¯g2[by deﬁnition of kernel]

=Im ¯

f2[Since the equation is exact]. Now, as ¯

is injective homomorphism we can write Im ¯

f2=

µ2. As a result, Ker¯π1=µ2. So, a fuzzy short

exact sequence can be formed

0→µ2→ν¯π1

−→ η1→0————-(3)

Since η1is a fuzzy p-poor module equation (3)

splits thus ∃¯

h:η1→νsuch that ¯π1o¯

Idη1. Hence by [17], we have ν=η1⊕µ2. In an

analogous way, another fuzzy short exact sequence

can be formed

0→µ1→ν¯π2

−→ η2→0 ——–(4)

to give ν=η2⊕µ1. Therefore µ1⊕η2∼

=µ2⊕

η1.

Lemma 4.10 For any ring R, TP(µP) = SS

Mod-FR where µPis fuzzy right R-module.

Proof. : The containment ⊇is obvious. Let µB∈

TP(µP) and µCis a fuzzy submodule of µB. Then

µB/µCis µB-projective. This implies µCis a fuzzy

direct summand of µB. Hence µBis fuzzy semisim-

ple.

Lemma 4.11 Let µMbe a fuzzy p-poor module.

Then for every νN,µM⊕νNis fuzzy p-poor.

Proof. : Let νNbe in Mod-FR and µM⊕νNis ηT-

projective. Then µMis ηT-projective. Since µMis

fuzzy p-poor, ηTmust be fuzzy semisimple. Thus,

µM⊕νNis fuzzy p-poor.

Lemma 4.12 If µM⊕νNis fuzzy p-poor and

µMis fuzzy projective then νNis fuzzy p-poor.

Proof. : Let νNis ηT-projective. Then µM⊕νNis

ηT-projective. Hence ηTis a fuzzy semisimple.

Lemma 4.13 For every ring R the following

are equivalent:

(i) R is fuzzy semisimple artinian.

(ii) Every fuzzy module µMis p-poor.

(iii) There exists a fuzzy projective p-poor R-

module.

Proof. : Let µMbelong to Mod-FR. Then SS Mod-

FR ⊆P(µM)⊆Mod-FR = SS Mod-FR. Hence (i)

⇒(ii). Also (ii) ⇒(iii) is clear. Assuming (iii) we

can have µMas a fuzzy projective p-poor module.

Thus, SS Mod-FR = P(µM) = Mod-FR which im-

plies (i).

Proposition 4.14 For every ring R the follow-

ing are equivalent:

(i) R is fuzzy semisimple artinian.

(ii) All fuzzy p-poor right(left)R-modules are fuzzy

semisimple.

(iii) Non zero direct summmands of fuzzy p-poor

right(left) R-modules are fuzzy p-poor.

Proof. : If R is fuzzy semisimple Artinian then (ii)

and (iii) holds good. If (ii) or (iii) holds true then

every fuzzy module is fuzzy p-poor, since a fuzzy

p-poor module exists and a direct sum of any fuzzy

module with a fuzzy p-poor module is again a p-

poor [lemma 4.11]. Thus, R is a fuzzy semisimple

Artinian [lemma 4.13].

Proposition 4.15 If HomR(µM, µA) = 0 then

µAbelongs to projectivity domain of µM.

Proof. : If HomR(µM, µA) = 0 then given any sur-

jective homomorphism ¯g:µC→µAand if we sup-

pose ¯

h:µM→µCbe a zero mapping then ¯g¯

h= 0.

This implies µAbelongs to the projectivity domain

of µM.

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5 Future Scope

The behavior of fuzzy projective modules in vari-

ous rings, such as the Laurent polynomial, QF and

Artinian Rings can always be investigated to add

another dimension to section 3. With the help of

[7], one may always try to apply the concepts pro-

duced during this study to the research stated in

[1] of fuzzy semirings, and can also try to exclude

fuzzy projective modules using fuzzy projective

semi modules supplied in [21]. Section 4 of current

study act as a necessary catalyst to ponder one

to choose an alternate perspective of fuzzy projec-

tivity, namely fuzzy subprojectively poor modules

whose domain contains precisely fuzzy projective

modules only. In the same vein, fuzzy sub injec-

tively poor modules can be studied to give fuzzy

dimension to the research mentioned in [5] and [20].

Also, the research mentioned in [3, 4, 8, 11] and

[12] can be extended using the current cancellation

study on fuzzy projective modules. In addition,

Schanuel’s Lemma, which is being explored in a

fuzzy setting is useful in giving a fresh and fuzzy

direction to many vital classical notions, like the

uniqueness of syzygy modules of a module up to

free summands and uniqueness of cosyzygy mod-

ules of a module up to injective summands.

6 Conclusion

The concept of cancellation of the fuzzy module

over the polynomial ring is initiated during this

study, along with which the necessary and suﬃ-

cient condition for the fuzzy projective module to

conciliate cancellation is discussed. We have also

introduced the concept of fuzzy p-poor modules,

which has proven to be a viable alternative to em-

ploy fuzzy projective modules during the equiva-

lence of fuzzy modules. To make it reader-friendly,

the study done in this paper is diagrammatically

summarized below:

Fig.6 Summarizing the types of Fuzzy Modules

studied in the paper

NOTE FOR FIGURE 6

(i) In deﬁnition 2.2 the fuzzy R-module µPcan

also, be called µA-projective or projective relative

to µA.

(ii) Projectivity domain of a module µPis the set

of all µA’s such that µPis µA-projective.

References

[1] Ahsan J., Saifullah K. and Khan F. M.,

“Fuzzy Semirings”, Fuzzy Sets and Systems

60 (3), 309-320, 1993.

[2] Anderson D. D and Anderson D. F., “Some

Remarks on Cancellation Ideals”, Math,

Japonica 29 (6), 1984.

[3] Bothaynah N. S., Khalaf H. Y. and Mahmood

L. S., “Purely and Weakly Purely Cancella-

tion Modules”, American Journal of Mathe-

matics and Statitics, 4 (4), 186-190, 2014.

[4] Bothaynah N. S and Khalaf H. Y., “Purely

Cancellation Fuzzy Module”, International

Journal of Advanced Scientiﬁc and Technical

Research, 4 (6), 2016.

[5] Fuchs L. and Loonstra F., “On the cancella-

tion of modules in direct sums over Dedekind

domains”, Indag. Math., 33 , 163-169, 1971.

[6] Gilmer R. W., “The cancellation law for ideals

in a commutative ring”. Canad. J. Math., 17,

281-287, 1964.

[7] Golan J.S., “The theory of semirings with

applications in mathematics and theoreti-

cal computer science”, Pitman Monographs

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Volume 21, 2022

and surveys in pure and applied Maths., no

54(Longman), New York, 1992.

[8] Hadi M. A., Abd M. “Cancellation and weakly

cancellation fuzzy modules”. Information and

control, 37 (4), 2011.

[9] Isaac P., “Studies in fuzzy commutative alge-

bra”, Ph.D thesis, 2004.

[10] Kaplansky I., “Fields and rings(Second Edi-

tion)”, Chicago Lectures In Mathematical Se-

ries, 165-168, 1972.

[11] Khalaf H. Y., “Restricted Cancellation and

Weakly Restricted Cancellation Fuzzy mod-

ules”, 2015.

[12] Khalaf H. Y. and Rashid H. G., “Fully and

Naturally Cancellation Fuzzy Module”, Inter-

national Journal of Applied Mathematics and

statistical sciences, 5 (2), 2016.

[13] Kumar R., Bhambri S. K., Pratibha, “Fuzzy

submodules: some analogues and deviations”,

Fuzzy sets and systems, 70 (1), 125-130, 1995.

[14] Liu H. X., “Fuzzy projective modules and ten-

sor products in fuzzy module categories”, Ira-

nian Journal of Fuzzy Systems, 11 (2), 89-101,

2014.

[15] Mahmood L. S., Bothaynah N. Shihab and

Tahir S. Rasheed, “Relatively cancellation

modules”, Journal of Al-Nahrain University,

13 (2), 175-182, 2010.

[16] Mijbass A. S., “On Cancellation Modules”,

M.Sc, Thesis, University of Baghdad, 1992.

[17] Mishra R. K, Kumar S. D, Behra S., “On

Projective Modules and Computation of Di-

mension of a Module over Laurent Polynomial

Ring, International Scholarly Research Net-

work, ISRN Algebra, Volume 2011.

[18] Naoum A. G and Mijbass A. S., “Weak Can-

cellation Modules”, Kyungpook Math. J. 37,

73-82, 1997.

[19] Permouth S. R. Lopez and Simental J. E.,

“Characterizing rings in terms of the extent of

injectivity and projectivity of their modules”,

J. Algebra 362, 56-69, 2012.

[20] Sahni A. K., Pandey J. T., Mishra R. K and

Sinha V. K., “Fuzzy Projective-Injective Mod-

ules and Their Evenness Over Semi-Simple

Rings”, Turkish Journal of Computer and

Mathematics Education, 12 (6), 3624-3634,

2021.

[21] Sahni A. K., Pandey J. T., Mishra R. K. and

Sinha V. K., On fuzzy proper exact sequences

and fuzzy projective semi modules over semir-

ings, WSEAS Transactions on Mathematics,

20, 700-711, 2021.

[22] Swan R. G., “A Cancellation Theorem

for Projective Modules in the Metastable

Range”, Inventiones Math, Springer Verlag,

27, 23–43,1974.

[23] Zahedi M. M. , Ameri R., “On fuzzy projec-

tive and injective modules”, J. Fuzzy Math, 3

(1), 181-190, 1995.

[24] Zhang H., Tong W., “The Cancellation Prop-

erty Of Projective Modules”, Algebra Collo-

quium, 13 (4), 617-622, 2006.

[25] Zinna Md Ali., “Cancellation of projective

modules over polynomial extensions over a

two-dimensional ring.” Journal of Pure and

Applied Algebra, 223 (2), 783-793, 2019.

Contribution of individual authors to the

creation of a scientiﬁc article (ghostwriting

policy)

Author Contributions:

Amarjit kaur sahni formulated the manuscript af-

ter the necessary literature review. Constructed

the examples, lemmas, theorems are given and de-

veloped the concept of Schanuel’s Lemma using

fuzzy p-poor modules.

Jayanti Tripathi Pandey suggested bridging the

gap between fuzzy module cancellation and fuzzy

projective module cancellation. In addition, the

notion of cancellation over polynomial rings and

ring R was advised to be studied. She also reviewed

the manuscript’s overall structure by giving helpful

suggestions.

Ratnesh Kumar Mishra proposed the idea of

cancellation on fuzzy projective modules.

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in a scientiﬁc article or scientiﬁc article it-

self

No fund

Creative Commons Attribution License

4.0 (Attribution 4.0 International , CC BY

4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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