# Radical Structures of Fuzzy Polynomial Ideals in a Ring

Authors: Hee Sik Kim; Chang Bum Kim; Keum Sook So
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

We investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study its properties. Given a fuzzy idealβ of R and a homomorphism f
:
R
→
R
′, we show that if f
x is the induced homomorphism of f, that is, f
x
(
∑
i
=
0
n
a
i
x
i
)
=
∑
i
=
0
n
f
(
a
i
)
x
i, then f
x
-
1
[
(
β
)
x
]
=
(
f
-
1
(
β
)
)
x.


---

## Body

## 1. Introduction

Zadeh [1] introduced the notion of a fuzzy subset A of a set X as a function from X into [
0,1
]. Rosenfeld [2] applied this concept to the theory of groupoids and groups. Liu [3] introduced and studied the notion of the fuzzy ideals of a ring. Following Liu, Mukherjee and Sen [4] defined and examined the fuzzy prime ideals of a ring. The concept of fuzzy ideals was applied to several algebras: B
N-algebras [5], B
L-algebras [6], semirings [7], and semigroups [8]. Ersoy et al. [9] applied the concept of intuitionistic fuzzy soft sets to rings, and Shah et al. [10] discussed intuitionistic fuzzy normal subrings over a nonassociative ring. Prajapati [11] investigated residual of ideals of an L-ring. Dheena and Mohanraj [12] obtained a condition for a fuzzy small right ideal to be fuzzy small prime right ideal.The present authors [13] introduced the notion of a fuzzy polynomial ideal α
x of a polynomial ring R
[
x
] induced by a fuzzy ideal α of a ring R and obtained an isomorphism theorem of a ring of fuzzy cosets of α
x. It was shown that a fuzzy ideal α of a ring is fuzzy prime if and only if α
x is a fuzzy prime ideal of R
[
x
]. Moreover, we showed that if α
x is a fuzzy maximal ideal of R
[
x
], then α is a fuzzy maximal ideal of R.In this paper we investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study their properties.


## 2. Preliminaries

In this section, we review some definitions which will be used in the later section. Throughout this paper unless stated otherwise all rings are commutative rings with identity.Definition 1 (see [3]).
Afuzzy ideal of a ring R is a function α
:
R
→
[
0,1
] satisfying the following axioms:(1)
α
(
x
+
y
)
≥
min
⁡
{
α
(
x
)
,
α
(
y
)
}.
(2)
α
(
x
y
)
≥
max
⁡
{
α
(
x
)
,
α
(
y
)
}.
(3)
α
(
-
x
)
=
α
(
x
)for any x
,
y
∈
R.Definition 2 (see [2]).
Letf
:
R
→
S be a homomorphism of rings and let β be a fuzzy subset of S. We define a fuzzy subset f
-
1
β of R by f
-
1
β
(
x
)
≔
β
(
f
(
x
)
) for all x
∈
R.Definition 3 (see [2]).
Letf
:
R
→
S be a homomorphism of rings and let α be a fuzzy subset of R. We define a fuzzy subset f
(
α
) of S by(1)
f
α
y
≔
sup
⁡
α
t
∣
t
∈
R
,
f
t
=
y
if
f
-
1
y
≠
∅
,
0
if
f
-
1
y
=
∅
.Definition 4 (see [2]).
LetR and S be any sets and let f
:
R
→
S be a function. A fuzzy subset α of R is called an f-invariant if f
(
x
)
=
f
(
y
) implies α
(
x
)
=
α
(
y
), where x
,
y
∈
R.Zadeh [1] defined the following notions. The union of two fuzzy subsets α and β of a set S, denoted by α
∪
β, is a fuzzy subset of S defined by(2)
α
∪
β
x
≔
max
⁡
α
x
,
β
xfor all x
∈
S.The intersection ofα and β, symbolized by α
∩
β, is a fuzzy subset of S, defined by(3)
α
∩
β
x
≔
min
⁡
α
x
,
β
xfor all x
∈
S.Theorem 5 (see [13]).
Letα
:
R
→
[
0,1
] be a fuzzy ideal of a ring R and let f
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n be a polynomial in R
[
x
]. Define a fuzzy set α
x
:
R
[
x
]
→
[
0,1
] by α
x
(
f
(
x
)
)
≔
min
i
⁡
α
a
i
∣
a
i
’s are coefficients of
f
x. Then α
x is a fuzzy ideal of R
[
x
].The fuzzy idealα
x discussed in Theorem 5 is called thefuzzy polynomial ideal [13] of R
[
x
] induced by a fuzzy ideal α.Theorem 6 (see [13]).
Letα
:
R
→
[
0,1
] be a fuzzy ideal of a ring R. Then α is a fuzzy prime ideal of R if and only if α
x is a fuzzy prime ideal of R
[
x
].Notation 1.
Letα
:
R
→
[
0,1
] be a fuzzy subset of a set R. We denote a level set α
∗ by α
∗
≔
a
∈
R
∣
α
a
=
α
0, and we know that α
(
0
)
≧
α
(
x
) for all x
∈
R. The set of all polynomials f
(
x
)
=
∑
i
=
0
m
a
i
x
i
∈
R
[
x
] whose α’s values α
(
a
i
) are equal to α
(
0
) for all i
=
0,1
,
…
,
n, is denoted by α
∗
[
x
].Theorem 7 (see [13]).
Ifα and β are fuzzy ideals of a ring R, then(i)
(
α
∩
β
)
x
=
α
x
∩
β
x,
(ii)
α
x
∪
β
x
⊆
(
α
∪
β
)
x
.Letf
:
R
→
R
′ be a homomorphism of rings. A map f
x
:
R
[
x
]
→
R
′
[
x
] defined by f
x
(
a
0
+
a
1
x
+
⋯
+
a
n
x
n
)
≔
f
(
a
0
)
+
f
(
a
1
)
x
+
⋯
+
f
(
a
n
)
x
n is obviously a ring homomorphism, and we call it aninduced homomorphism [13] by f.Theorem 8 (see [13]).
Letf
:
R
→
R
′ be an epimorphism of rings and let f
x be an induced homomorphism of f. If α is an f-invariant fuzzy ideal of R, then (
f
(
α
)
)
x
=
f
x
(
α
x
)
.


## 3. Fuzzy Polynomial Ideals

In this section, we study some relations between the radical of the fuzzy polynomial induced by a fuzzy ideal and the radical of a fuzzy ideal of a ring.A fuzzy idealα
:
R
→
[
0,1
] of a ring R is called afuzzy prime ideal [14] of R if α
∗ is a prime ideal of R. A fuzzy set α
:
R
→
[
0,1
], defined as α
(
a
)
:
=
⋁
α
a
n
∣
n
>
0, is called afuzzy nil radical [15] of α.Theorem 9 (see [15]).
Ifα
:
R
→
[
0,1
] is a fuzzy ideal of R, then the fuzzy set α is a fuzzy ideal of R.Lemma 10 (see [15]).
Ifα and β are fuzzy ideals of R, then α
∩
β
=
α
∩
β.Lemma 11.
Ifα and β are fuzzy ideals of R, then α
∪
β
=
α
∪
β.Proof.
Ifa is an element of R, then(4)
α
∪
β
a
=
⋁
α
∪
β
a
n
∣
n
>
0
=
⋁
max
⁡
α
a
n
,
β
a
n
∣
n
>
0
=
max
⁡
⋁
α
a
n
∣
n
>
0
,
⋁
β
a
n
∣
n
>
0
=
max
⁡
α
a
,
β
a
=
α
∪
β
a
.This proves that α
∪
β
=
α
∪
β.Sinceα
x is a fuzzy ideal of a polynomial ring R
[
x
] by Theorem 5, the fuzzy set α
x
:
R
[
x
]
→
[
0,1
] is the fuzzy nil radical of α
x. The following theorem gives that the two fuzzy nil radicals have the same value.Theorem 12.
Ifα
:
R
→
[
0,1
] is a fuzzy ideal of R, then(5)
α
x
x
=
α
x
.Proof.
Letf
(
x
)
≔
∑
i
=
0
m
‍
∈
R
[
x
] be any element of R
[
x
]. Then, by Theorem 5, we have α
x
(
a
j
n
)
=
α
x
(
a
j
n
+
0
x
+
0
x
2
+
⋯
+
0
x
m
)
=
min
⁡
α
a
j
n
,
α
0
,
…
,
α
0
=
α
(
a
j
n
). Since α
x is a fuzzy ideal of R
[
x
], we obtain(6)
α
x
x
f
x
=
min
i
=
0
m
⁡
α
x
a
i
=
min
i
=
0
m
⁡
⋁
α
x
a
i
n
∣
n
>
0
=
min
i
=
0
m
⁡
⋁
α
a
i
n
∣
n
>
0
=
min
i
=
0
m
⁡
α
a
i
=
α
x
f
x
.This proves that (
α
)
x
=
(
α
x
)
x.Theorem 13.
Ifα and β are fuzzy ideals of R, then(7)
α
∩
β
x
=
α
x
∩
β
x
.Proof.
Ifα and β are fuzzy ideals of R, then α
x and β
x are fuzzy ideals of R
[
x
] by Theorem 5. It follows from Theorems 12 and 7(i) and Lemma 10 that(8)
α
∩
β
x
=
α
∩
β
x
x
T
h
e
o
r
e
m
12
=
α
x
∩
β
x
x
T
h
e
o
r
e
m
7
i
=
α
x
∩
β
x
x
L
e
m
m
a
10
=
α
x
x
∩
β
x
x
T
h
e
o
r
e
m
7
i
=
α
x
∩
β
x
T
h
e
o
r
e
m
12proving the theorem.Theorem 14.
Ifα and β are fuzzy ideals of R, then(9)
α
x
∪
β
x
⊆
α
∪
β
x
.Proof.
Sinceα
x and β
x are fuzzy ideals of R
[
x
] by Theorem 5, we obtain(10)
α
x
∪
β
x
=
α
x
x
∪
β
x
x
T
h
e
o
r
e
m
12
⊆
α
x
∪
β
x
x
T
h
e
o
r
e
m
7
i
i
=
α
x
∪
β
x
x
L
e
m
m
a
11
⊆
α
∪
β
x
x
T
h
e
o
r
e
m
7
i
i
=
α
∪
β
x
T
h
e
o
r
e
m
12
.This proves that (
α
)
x
∪
(
β
)
x
⊆
(
α
∪
β
)
x.Theorem 15.
Letβ be a fuzzy ideal of R and let f
:
R
→
R
′ be a homomorphism of rings. If f
x is the induced homomorphism of f, that is, f
x
(
∑
i
=
0
n
a
i
x
i
)
=
∑
i
=
0
n
f
(
a
i
)
x
i, then(11)
f
x
-
1
β
x
=
f
-
1
β
x
.Proof.
Given a polynomialg
(
x
)
=
b
0
+
b
1
x
+
⋯
+
b
m
x
m
∈
R
[
x
], we have(12)
f
-
1
β
x
g
x
=
min
⁡
f
-
1
β
b
0
,
f
-
1
β
b
1
,
…
,
f
-
1
β
b
m
=
min
⁡
⋁
f
-
1
β
b
0
n
∣
n
>
0
,
⋁
f
-
1
β
b
1
n
∣
n
>
0
,
…
,
⋁
f
-
1
β
b
m
n
∣
n
>
0
=
min
⁡
⋁
β
f
b
0
n
∣
n
>
0
,
⋁
β
f
b
1
n
∣
n
>
0
,
…
,
⋁
β
f
b
m
n
∣
n
>
0
=
min
⁡
β
f
b
0
,
β
f
b
1
,
…
,
β
f
b
m
=
β
x
f
x
g
x
=
f
x
-
1
β
x
g
x
.This proves that f
x
-
1
[
(
β
)
x
]
=
(
f
-
1
(
β
)
)
x.Proposition 16 (see [15]).
Letf
:
R
→
R
′ be a ring epimorphism from R onto R
′, and let α
:
R
→
[
0,1
] be a fuzzy ideal of R. If α is constant on Ker
⁡
f, then f
(
α
)
=
f
(
α
).Theorem 17.
Letf
:
R
→
R
′ be a homomorphism of rings and let f
x be the induced homomorphism of f. If a fuzzy ideal α of R is constant on Ker
⁡
f, then the fuzzy polynomial ideal α
x is constant on Ker
⁡
f
x.Proof.
Letα
(
a
)
=
k
0 for all a
∈
Ker
⁡
f and let g
(
x
)
=
b
0
+
b
1
x
+
⋯
+
b
m
x
m be any element of Ker
⁡
f
x. Then 0
=
f
x
(
g
(
x
)
)
=
f
(
b
0
)
+
f
(
b
1
)
x
+
⋯
+
f
(
b
m
)
x
m. It follows that f
(
b
i
)
=
0 for all i
=
0,1
,
…
,
m. Hence b
i
∈
Ker
⁡
f for all i
=
0,1
,
…
,
m; that is, α
(
b
i
)
=
k
0 for all i
=
0,1
,
…
,
m. This shows that α
x
(
g
(
x
)
)
=
m
i
n
{
α
(
b
0
)
,
α
(
b
1
)
,
…
,
α
(
b
m
)
}
=
k
0, proving the theorem.Corollary 18.
Letf
:
R
→
R
′ be an epimorphism of rings and let f
x be the induced homomorphism of f. If an f-invariant fuzzy ideal α of R is constant on Ker
⁡
f, then(13)
f
x
α
x
=
f
α
x
.Proof.
It follows from Proposition16 and Theorem 8 that f
x
(
α
x
)
=
f
x
(
α
x
)
=
(
f
(
α
)
)
x.Theorem 19.
Letα be a fuzzy ideal of R and let α
x be its fuzzy polynomial ideal of R
[
x
]. If β is a fuzzy prime ideal of R
[
x
] such that α
x
⫅
β, then there exists a fuzzy prime ideal α
0 of R such that (
α
0
)
∗
=
β
∗
∩
R and α
⫅
α
0.Proof.
Sinceβ is a fuzzy prime ideal of R
[
x
], β
∗ is a prime ideal of R
[
x
]. If we define γ
:
=
β
∗
∩
R, then it is easy to show that γ is a prime ideal of R. Define a fuzzy subset α
0
:
R
→
[
0,1
] by(14)
α
0
a
=
β
0
if
a
∈
γ
,
β
a
if
a
∉
γ
.Then, by routine calculations, we show that α
0 is a fuzzy ideal of R satisfying (
α
0
)
∗
=
γ. We claim that α
⫅
α
0. Given a
∈
R, if a
∈
γ, then α
(
a
)
=
α
x
(
a
)
≤
β
(
a
)
≤
β
(
0
)
=
α
0
(
a
). If a
∉
γ, then α
(
a
)
=
α
x
(
a
)
≤
β
(
a
)
=
α
0
(
a
). Since γ is a prime ideal of R, (
α
0
)
∗ is a prime ideal of R. This shows that α
0 is a fuzzy prime ideal of R, proving the theorem.Definition 20 (see [15]).
Letα
:
R
→
[
0,1
] be a fuzzy ideal of R. The fuzzy ideal r
(
α
) defined by(15)
r
α
≔
⋂
β
∣
α
⫅
β
,
β
:  a fuzzy prime ideal of
Ris called theprime fuzzy radical of α.Theorem 21.
Letα be a fuzzy ideal of R and let α
x be its fuzzy polynomial ideal of R
[
x
]. Then(16)
r
α
x
⊆
r
α
x
.Proof.
By Theorem6, β
i is a fuzzy prime ideal of R with α
⊆
β
i if and only if (
β
i
)
x is a fuzzy prime ideal of R
[
x
] with α
x
⊆
(
β
i
)
x. It follows from Theorem 7(i) that(17)
r
α
x
=
⋂
β
i
∣
α
⊆
β
i
,
β
i
is a fuzzy prime ideal of
R
x
=
⋂
β
i
x
∣
α
⊆
β
i
,
β
i
is a fuzzy prime ideal of
R
=
⋂
β
i
x
∣
α
x
⊆
β
i
x
,
β
i
x
is a fuzzy prime ideal of
R
x
⊇
⋂
γ
i
∣
α
x
⊆
γ
i
,
γ
i
is a fuzzy prime ideal of
R
x
=
r
α
x
,proving the theorem.Notation 2.
Letα be a fuzzy ideal of R and let α
x be its fuzzy polynomial ideal of R
[
x
]. We denote F
P
I
(
α
) by(18)
FPI
α
≔
β
∣
α
⊆
β
,
β
is a fuzzy prime ideal of
Rand F
P
I
(
α
x
) by(19)
FPI
α
x
≔
γ
∣
α
x
⊆
γ
,
γ
is a fuzzy prime ideal of
R
x
.Theorem 22.
Letα be a fuzzy ideal of R and let α
x be its fuzzy polynomial ideal of R
[
x
]. Then a map ϕ
:
F
P
I
(
α
)
→
F
P
I
(
α
x
) defined by ϕ
(
β
)
≔
β
x is one-one.Proof.
Ifβ
,
γ
∈
F
P
I
(
α
) such that ϕ
(
β
)
=
ϕ
(
γ
), then β
x
=
γ
x. It follows that β
x
(
a
)
=
γ
x
(
a
) for all a
∈
R, and hence β
(
a
)
=
γ
(
a
) for all a
∈
R, proving that β
=
γ. Hence ϕ is one-one.Corollary 23.
Letα be a fuzzy ideal of R and let α
x be its fuzzy polynomial ideal of R
[
x
]. If the map ϕ defined in Theorem 22 is an onto map, then(20)
r
α
x
=
r
α
x
.Proof.
Ifβ is any element of F
P
I
(
α
x
), then there exists γ
∈
F
P
I
(
α
) such that γ
x
=
ϕ
(
γ
)
=
β with α
⊆
γ. Thus (
r
(
α
)
)
x
=
r
(
α
x
). This shows that the reverse inclusion in Theorem 21 holds.Example 24.
LetZ be set of all integers. Let(21)
α
x
≔
1
if
x
∈
2
Z
,
0
if
x
∉
2
Z
.Then α is a fuzzy prime ideal of Z, since α
∗
=
2
Z is a prime ideal of Z, and its induced polynomial ideal α
x is (22)
α
x
f
x
≔
1
if
f
x
∈
2
Z
x
,
0
if
f
x
∉
2
Z
x
.By Theorem 6, the fuzzy polynomial ideal α
x induced by α is a fuzzy prime ideal of Z
[
x
]. Hence (
r
(
α
)
)
x
=
α
x
=
r
(
α
x
).



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