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Fuzzy prime ideals redefined
Journal of Inequalities and Applications volume 2012, Article number: 203 (2012)
Abstract
In order to generalize the notions of a (\in ,\in \vee q)fuzzy subring and various (\in ,\in \vee q)fuzzy ideals of a ring, a (\lambda ,\mu )fuzzy subring and a (\lambda ,\mu )fuzzy ideal of a ring are defined. The concepts of (\lambda ,\mu )fuzzy semiprime, prime, semiprimary and primary ideals are introduced, and the characterizations of such fuzzy ideals are obtained based on a (\lambda ,\mu )cut set.
MSC:16L99, 03E72.
1 Introduction
The concept of a fuzzy set introduced by Zadeh [1] was applied to the group theory by Rosenfeld [2] and the ring theory by Liu [3]. Since then, many scholars have studied the theories of fuzzy subrings and various fuzzy prime ideals [4–6]. It is worth pointing out that Bhakat and Das introduced the concept of an (\alpha ,\beta )fuzzy subgroup by using the ‘belongs to’ relation and ‘quasicoincident with’ relation between a fuzzy point and a fuzzy subset, and gave the concepts of an (\in ,\in \vee q)fuzzy subgroup and an (\in ,\in \vee q)fuzzy subring [7, 8]. It is well known that a fuzzy subset A of a group G is a Rosenfeld’s fuzzy subgroup if and only if {A}_{t}=\{x\in G\mid A(x)\u2a7et\} is a subgroup of G for all t\in (0,1] (for our convenience, here ∅ is regarded as a subgroup of G). Similarly, A is an (\in ,\in \vee q)fuzzy subgroup if and only if {A}_{t} is a subgroup of G for all t\in (0,0.5]. A corresponding result should be considered when {A}_{t} is a subgroup of G for all t\in (a,b], where (a,b] is an arbitrary subinterval of [0,1]. Motivated by the above problem, Yuan et al. [9] introduced a fuzzy subgroup with the thresholds of a group. In order to generalize the concepts of an (\in ,\in \vee q)fuzzy subring and an (\in ,\in \vee q)fuzzy ideal of a ring, Yao [10] introduced the notions of a (\lambda ,\mu )fuzzy subring and a (\lambda ,\mu )fuzzy ideal and discussed their fundamental properties. In this paper, we will introduce the concepts of (\lambda ,\mu )fuzzy prime, fuzzy semiprime, fuzzy primary and fuzzy semiprimary ideals of a ring.
2 (\lambda ,\mu )fuzzy ideal
Let X be a nonempty set. By a fuzzy subset A of X, we mean a map from X to the interval [0,1], A:X\to [0,1]. If A is a fuzzy subset of X and t\in [0,1], then the cut set {A}_{t} and the open cut set {A}_{\u3008t\u3009} of A are defined as follows:
First, we recall some definitions and results for the sake of completeness.
Definition 1 Let x\in X, t\in (0,1]. A fuzzy subset A of X of the form
is said to be a fuzzy point with support x and value t and is denoted by {x}_{t}.
Definition 2 [11]
A fuzzy point {x}_{t} is said to belong to (resp. be quasicoincident with) a fuzzy subset A, written as {x}_{t}\in A (resp. {x}_{t}qA), if
{x}_{t}\in A or {x}_{t}qA will be denoted by {x}_{t}\in \vee qA.
In the following discussions, R always stands for an associate ring, λ and μ are constant numbers such that 0\le \lambda <\mu \le 1, and N denotes the set of all positive integers.
Definition 3 [8]
A fuzzy subset A of R is said to be an (\in ,\in \vee q)fuzzy subring of R if for all x,y\in R and t,r\in (0,1],

(1)
{x}_{t},{y}_{r}\in A\u27f9{(x+y)}_{t\wedge r}\in \vee qA,

(2)
{x}_{t}\in A\u27f9{(x)}_{t}\in \vee qA,

(3)
{x}_{t},{y}_{r}\in A\u27f9{(xy)}_{t\wedge r}\in \vee qA.
Definition 4 [8]
A fuzzy subset A of R is said to be an (\in ,\in \vee q)fuzzy ideal of R if

(1)
A is an (\in ,\in \vee q)fuzzy subring of R,

(2)
{x}_{t}\in A,y\in R\u27f9{(xy)}_{t},{(yx)}_{t}\in \vee qA,\mathrm{\forall}t\in (0,1].
According to Definition 3 and Definition 4, we have that a fuzzy subset A of R is a (\lambda ,\mu )fuzzy subring (ideal) of R if and only if for all x,y\in R,

(1)
A(xy)\ge A(x)\wedge A(y)\wedge 0.5,

(2)
A(xy)\ge A(x)\wedge A(y)\wedge 0.5 ((2)^{′} A(xy)\ge (A(x)\vee A(y))\wedge 0.5).
In order to give more general concepts of a fuzzy subring and a fuzzy ideal of R, we introduce the following definitions.
Definition 5 A fuzzy subset A of R is said to be a (\lambda ,\mu )fuzzy addition subgroup of R if for all x,y\in R,
Clearly, a fuzzy subset A of R is a (\lambda ,\mu )fuzzy addition subgroup of R if and only if for all x,y\in R, A(xy)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu.
Definition 6 [10]
A fuzzy subset A of R is said to be a (\lambda ,\mu )fuzzy subring of R if for all x,y\in R,

(1)
A(xy)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu,

(2)
A(xy)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu.
Definition 7 [10]
A fuzzy subset A of R is said to be a (\lambda ,\mu )fuzzy left ideal (resp. fuzzy right ideal) of R if for all x,y\in R,

(1)
A(xy)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu,

(2)
A(xy)\vee \lambda \ge A(y)\wedge \mu (resp. A(xy)\vee \lambda \ge A(x)\wedge \mu).
A is said to be a (\lambda ,\mu )fuzzy ideal of R if it is both a (\lambda ,\mu )fuzzy left ideal and a (\lambda ,\mu )fuzzy right ideal of R.
According to the above definitions, a (\lambda ,\mu )fuzzy left ideal or a (\lambda ,\mu )fuzzy right ideal of R must be a (\lambda ,\mu )fuzzy subring. A fuzzy subset A of R is a (\lambda ,\mu )fuzzy ideal of R if and only if for all x,y\in R,

(1)
A(xy)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu,

(2)
A(xy)\vee \lambda \ge (A(x)\vee A(y))\wedge \mu.
Obviously, an (\in ,\in \vee q)fuzzy subring (fuzzy ideal) of R is a (\lambda ,\mu )fuzzy subring (fuzzy ideal) of R with \lambda =0 and \mu =0.5.
The following theorem is obvious.
Theorem 1 Let A, B be (\lambda ,\mu )fuzzy left ideals (fuzzy right ideals, fuzzy ideals, fuzzy subrings) of R. Then A\cap B is also a fuzzy left ideal (fuzzy right ideal, fuzzy ideal, fuzzy subring) of R.
Theorem 2 Let A, B be (\lambda ,\mu )fuzzy left ideals (fuzzy right ideals, fuzzy ideals) of R. Then A+B is also a (\lambda ,\mu )fuzzy left ideal (fuzzy right ideal, fuzzy ideal) of R, where
Proof We only prove the case of a (\lambda ,\mu )fuzzy left ideal.
For all x,y\in R, we have
So, A+B is a (\lambda ,\mu )fuzzy left ideal of R.
Let A, B be fuzzy subsets of R. Then the fuzzy subset A\odot B is defined as follows: \mathrm{\forall}x\in R,
□
Theorem 3 Let A be a (\lambda ,\mu )fuzzy left ideal, and let B be a fuzzy subset of R. Then A\odot B is a (\lambda ,\mu )fuzzy left ideal of R.
Proof For all {z}_{1},{z}_{2}\in R, we have
So, A\odot B is a (\in ,\in \vee q)fuzzy left ideal of R. □
Similarly, we have the following theorem.
Theorem 4 Let A be a fuzzy subset, and let B be a (\lambda ,\mu )fuzzy right ideal of R. Then A\odot B is a (\lambda ,\mu )fuzzy right ideal of R.
The following theorem is an immediate consequence of Theorem 3 and Theorem 4.
Theorem 5 Let A be a (\lambda ,\mu )fuzzy left ideal, and let B be a (\lambda ,\mu )fuzzy right ideal of R. Then A\odot B is a (\lambda ,\mu )fuzzy ideal of R.
One of the most common methods of studying a fuzzy subring and a fuzzy ideal is by using their cut sets. Now we give the relation between a (\lambda ,\mu )fuzzy subring (fuzzy ideal) with its cut set or open cut set.
Theorem 6 [10]
A fuzzy subset A of R is a (\lambda ,\mu )fuzzy subring (fuzzy ideal) of R if and only if for all t\in (\lambda ,\mu ], {A}_{t} is a subring (ideal) of R or {A}_{t}=\mathrm{\varnothing}.
Theorem 7 A fuzzy subset A of R is a (\lambda ,\mu )fuzzy subring (fuzzy ideal) of R if and only if for all t\in [\lambda ,\mu ), {A}_{\u3008t\u3009} is a subring (ideal) of R or {A}_{\u3008t\u3009}=\mathrm{\varnothing}.
Proof We only prove the case of a (\lambda ,\mu )fuzzy subring.
Let A be a (\lambda ,\mu )fuzzy subring of R, and let t\in [\lambda ,\mu ). If x,y\in {A}_{\u3008t\u3009}, then A(x)\wedge A(y)>t, A(xy)\vee \lambda \u2a7eA(x)\wedge A(y)\wedge \mu >t. Considering \lambda \u2a7dt, we have A(xy)>t and xy\in {A}_{\u3008t\u3009}. Similarly, xy\in {A}_{\u3008t\u3009}. It follows that {A}_{\u3008t\u3009} is a subring of R.
Conversely, assume that {A}_{\u3008t\u3009} is a subring of R for all t\in [\lambda ,\mu ). If possible, let A({x}_{0}{y}_{0})\vee \lambda <A({x}_{0})\wedge A({y}_{0})\wedge \mu for some {x}_{0},{y}_{0}\in R. Put t=A({x}_{0}{y}_{0})\vee \lambda, then t\in [\lambda ,\mu ), and A({x}_{0}{y}_{0})\u2a7dt, A({x}_{0})\wedge A({y}_{0})>t. So, {x}_{0},{y}_{0}\in {A}_{\u3008t\u3009}, and {x}_{0}{y}_{0}\notin {A}_{\u3008t\u3009}. This is a contradiction to the fact that {A}_{\u3008t\u3009} is a subring of R. This shows that A(xy)\vee \lambda \u2a7eA(x)\wedge A(y)\wedge \mu holds for all x,y\in R.
Similarly, A(xy)\vee \lambda \u2a7eA(x)\wedge A(y)\wedge \mu, \mathrm{\forall}x,y\in R. That is, {A}_{\u3008t\u3009} is a subring of R. □
In the following theorems, it is shown that the homomorphism image (preimage) of a (\lambda ,\mu )fuzzy subring is also a (\lambda ,\mu )fuzzy subring. Similar result can be obtained for a (\lambda ,\mu )fuzzy ideal under some conditions.
Theorem 8 [10]
Let f:R\to {R}^{\mathrm{\prime}} be a homomorphism of rings. If A is a (\lambda ,\mu )fuzzy subring of R, then f(A) is a (\lambda ,\mu )fuzzy subring of {R}^{\mathrm{\prime}}. Particularly, if A is a (\lambda ,\mu )fuzzy ideal of R and f is onto, then f(A) is a (\lambda ,\mu )fuzzy ideal of {R}^{\mathrm{\prime}}, where
Theorem 9 [10]
Let f:R\to {R}^{\mathrm{\prime}} be a homomorphism of rings. If B is a (\lambda ,\mu )fuzzy subring (fuzzy ideal) of {R}^{\mathrm{\prime}}, then {f}^{1}(B) is a (\lambda ,\mu )fuzzy subring (fuzzy ideal) of R, where
3 (\lambda ,\mu )cut set
Based on the notion of an (\in ,\in \vee q)level subset defined in [12], we introduce the concept of a (\lambda ,\mu )cut set of a fuzzy subset. Let A be a fuzzy subset of a set X and t\in [0,1]. Then the subset {A}_{t}^{(\lambda ,\mu )} of X defined by
is said to be a (\lambda ,\mu )cut set of A. We denote {x}_{t}{\in}_{(\lambda ,\mu )}A if x\in {A}_{t}^{(\lambda ,\mu )}.
Obviously, if A is a fuzzy subset of X and t\in [0,1], then
Moreover, {x}_{t}\in \vee qA coincides with {x}_{t}{\in}_{(0,0.5)}A, and {x}_{t}\in \vee {q}_{k}A [12] coincides with {x}_{t}{\in}_{(0,\frac{k}{2})}A.
Lemma 1 Let A, B be fuzzy subsets of X. Then for all t\in [0,1],
Proof The proof is straightforward. □
Theorem 10 Let A, B be fuzzy subsets of X and t\in [0,1]. Then

(1)
{(A\cap B)}_{t}^{(\lambda ,\mu )}={A}_{t}^{(\lambda ,\mu )}\cap {B}_{t}^{(\lambda ,\mu )},

(2)
{(A\cup B)}_{t}^{(\lambda ,\mu )}={A}_{t}^{(\lambda ,\mu )}\cup {B}_{t}^{(\lambda ,\mu )}.
Proof
(1) Obviously, we have {(A\cap B)}_{t}^{(\lambda ,\mu )}\subseteq {A}_{t}^{(\lambda ,\mu )}\cap {B}_{t}^{(\lambda ,\mu )} from Lemma 1. If x\in {A}_{t}^{(\lambda ,\mu )}\cap {B}_{t}^{(\lambda ,\mu )}, then x\in {A}_{t}^{(\lambda ,\mu )} and x\in {B}_{t}^{(\lambda ,\mu )}. So, we have the following four cases.
Case 1, if A(x)\vee \lambda \u2a7et\wedge \mu and B(x)\vee \lambda \u2a7et\wedge \mu, then (A\cap B)(x)\vee \lambda \u2a7et\wedge \mu. So, x\in {(A\cap B)}_{t}^{(\lambda ,\mu )}.
Case 2, if A(x)>(2\mu t)\vee \lambda and B(x)>(2\mu t)\vee \lambda, then (A\cap B)(x)>(2\mu t)\vee \lambda. So, x\in {(A\cap B)}_{t}^{(\lambda ,\mu )}.
Case 3, if A(x)\vee \lambda \u2a7et\wedge \mu and B(x)>(2\mu t)\vee \lambda, then when t\u2a7d\mu, we have B(x)\vee \lambda >(2\mu t)\vee \lambda \u2a7e\mu \u2a7et\wedge \mu and hence (A\cap B)(x)\vee \lambda \u2a7et\wedge \mu. When t>\mu, we have A(x)\u2a7e\mu >(2\mu t)\vee \lambda and hence (A\cap B)(x)>(2\mu t)\vee \lambda. It means that x\in {(A\cap B)}_{t}^{(\lambda ,\mu )}.
Case 4, if A(x)>(2\mu t)\vee \lambda and B(x)\vee \lambda \u2a7et\wedge \mu, then we can also obtain that x\in {(A\cap B)}_{t}^{(\lambda ,\mu )} just as in Case 3.
Therefore, {(A\cap B)}_{t}^{(\lambda ,\mu )}={A}_{t}^{(\lambda ,\mu )}\cap {B}_{t}^{(\lambda ,\mu )}.
(2) Obviously, we have {(A\cup B)}_{t}^{(\lambda ,\mu )}\supseteq {A}_{t}^{(\lambda ,\mu )}\cup {B}_{t}^{(\lambda ,\mu )} from Lemma 1.
Let x\in {(A\cup B)}_{t}^{(\lambda ,\mu )}, then either A(x)\vee B(x)\vee \lambda \u2a7et\wedge \mu or A(x)\vee B(x)>(2\mu t)\vee \lambda. It means that A(x)\vee \lambda \u2a7et\wedge \mu, or B(x)\vee \lambda \u2a7et\wedge \mu, or A(x)>(2\mu t)\vee \lambda, or B(x)>(2\mu t)\vee \lambda. So x\in {A}_{t}^{(\lambda ,\mu )} or x\in {B}_{t}^{(\lambda ,\mu )}. That is, x\in {A}_{t}^{(\lambda ,\mu )}\cup {B}_{t}^{(\lambda ,\mu )}. Hence, {(A\cup B)}_{t}^{(\lambda ,\mu )}={A}_{t}^{(\lambda ,\mu )}\cup {B}_{t}^{(\lambda ,\mu )}. □
Theorem 11 Let A, B, C be fuzzy subsets of X and t\in [0,1]. Then

(1)
{[A\cap (B\cup C)]}_{t}^{(\lambda ,\mu )}={(A\cap B)}_{t}^{(\lambda ,\mu )}\cup {(A\cap C)}_{t}^{(\lambda ,\mu )},

(2)
{[A\cup (B\cap C)]}_{t}^{(\lambda ,\mu )}={(A\cup B)}_{t}^{(\lambda ,\mu )}\cap {(A\cup C)}_{t}^{(\lambda ,\mu )}.
Proof The proof can be obtained immediately from Theorem 10. □
Theorem 12 Let A be a (\lambda ,\mu )fuzzy subring (fuzzy ideal) of R. Then for all t\in [0,1], {A}_{t}^{(\lambda ,\mu )} is a subring (ideal) of R or {A}_{t}^{(\lambda ,\mu )}=\mathrm{\varnothing}.
Proof We only prove the case of a (\lambda ,\mu )fuzzy subring.
If t\u2a7d\lambda, then {A}_{t}^{(\lambda ,\mu )}=R. If \lambda <t\u2a7d\mu, then {A}_{t}^{(\lambda ,\mu )}={A}_{t} and {A}_{t} is a subring of R from Theorem 6. If t>\mu, then {A}_{t}^{(\lambda ,\mu )}={A}_{\u3008(2\mu t)\vee \lambda \u3009} and (2\mu t)\vee \lambda \in [\lambda ,\mu ). So, {A}_{t}^{(\lambda ,\mu )} is a subring of R from Theorem 7. □
Theorem 13 Let A be a fuzzy subset of R. If for all t\in (\lambda ,\mu ], {A}_{t}^{(\lambda ,\mu )} is a subring (ideal) of R or {A}_{t}^{(\lambda ,\mu )}=\mathrm{\varnothing}, then A is a (\lambda ,\mu )fuzzy subring (fuzzy ideal) of R.
Proof The proof can be obtained from Theorem 6. □
4 (\lambda ,\mu )fuzzy prime and fuzzy primary ideal
There are several deferent definitions of a fuzzy prime ideal and a fuzzy primary ideal of R. In this section, by a prime ideal S of R, we mean an ideal of R such that ab\in S\u27f9a\in S or b\in S.
Bhakat and Das [8] defined fuzzy prime, fuzzy semiprime, fuzzy primary and fuzzy semiprimary ideals in a ring which must be (\in ,\in \vee q)fuzzy ideals first.
Definition 8 [8]
An (\in ,\in \vee q)fuzzy ideal A of R is said to be

(1)
fuzzy semiprime if for all x\in R and t\in (0,1], {({x}^{2})}_{t}\in A\u27f9{x}_{t}\in \vee qA,

(2)
fuzzy prime if for all x,y\in R and t\in (0,1], {(xy)}_{t}\in A\u27f9{x}_{t}\in \vee qA or {y}_{t}\in \vee qA,

(3)
fuzzy semiprimary if for all x,y\in R and t\in (0,1], {(xy)}_{t}\in A\u27f9{x}_{t}^{m}\in \vee qA or {y}_{t}^{n}\in \vee qA for some m,n\in N,

(4)
fuzzy primary if for all x,y\in R and t\in (0,1], {(xy)}_{t}\in A\u27f9{x}_{t}\in \vee qA or {y}_{t}^{m}\in \vee qA for some m\in N.
In order to generalize these notions, we introduce (\lambda ,\mu )fuzzy prime, (\lambda ,\mu )fuzzy semiprime, (\lambda ,\mu )fuzzy primary and (\lambda ,\mu )fuzzy semiprimary ideals.
Definition 9 A (\lambda ,\mu )fuzzy ideal A of R is said to be

(1)
(\lambda ,\mu )fuzzy semiprime if for all x\in R and t\in (0,1], {({x}^{2})}_{t}\in A\u27f9{x}_{t}{\in}_{(\lambda ,\mu )}A,

(2)
(\lambda ,\mu )fuzzy prime if for all x,y\in R and t\in (0,1], {(xy)}_{t}\in A\u27f9{x}_{t}{\in}_{(\lambda ,\mu )}A or {y}_{t}{\in}_{(\lambda ,\mu )}A,

(3)
(\lambda ,\mu )fuzzy semiprimary if for all x,y\in R and t\in (0,1], {(xy)}_{t}\in A\u27f9{x}_{t}^{m}{\in}_{(\lambda ,\mu )}A or {y}_{t}^{n}{\in}_{(\lambda ,\mu )}A for some m,n\in N,

(4)
(\lambda ,\mu )fuzzy primary if for all x,y\in R and t\in (0,1], {(xy)}_{t}\in A\u27f9{x}_{t}{\in}_{(\lambda ,\mu )}A or {y}_{t}^{m}{\in}_{(\lambda ,\mu )}A for some m\in N.
In the following four theorems, we give the equivalence condition of a (\lambda ,\mu )fuzzy prime (semiprime, primary, semiprimary) ideal.
Theorem 14 A (\lambda ,\mu )fuzzy ideal A of R is (\lambda ,\mu )fuzzy prime if and only if for all x,y\in R,
Proof Let A be (\lambda ,\mu )fuzzy prime. If possible, let \lambda \vee A({x}_{0})\vee A({y}_{0})<A({x}_{0}{y}_{0})\wedge \mu for some {x}_{0},{y}_{0}\in R. Put t=A({x}_{0}{y}_{0})\wedge \mu, then A({x}_{0})\vee A({y}_{0})<t and {({x}_{0}{y}_{0})}_{t}\in A. So, \lambda \vee A({x}_{0})<t=t\wedge \mu and \lambda \vee A({y}_{0})<t=t\wedge \mu. From \lambda \vee A({x}_{0})<t\wedge \mu, we have A({x}_{0})\u2a7d\lambda \vee A({x}_{0})<t\u2a7d\mu \u2a7d2\mu t and hence {x}_{0}\notin {A}_{t}^{(\lambda ,\mu )}. Similarly, {y}_{0}\notin {A}_{t}^{(\lambda ,\mu )}. This is a contradiction. Therefore, for all x,y\in R, we have \lambda \vee A(x)\vee A(y)\u2a7eA(xy)\wedge \mu.
Conversely, if for all x,y\in R, \lambda \vee A(x)\vee A(y)\u2a7eA(xy)\wedge \mu and {(xy)}_{t}\in A, then \lambda \vee A(x)\vee A(y)\u2a7et\wedge \mu. So, \lambda \vee A(x)\u2a7et\wedge \mu or \lambda \vee A(y)\u2a7et\wedge \mu. That is, x{\in}_{(\lambda ,\mu )}A or y{\in}_{(\lambda ,\mu )}A. Hence, A is (\lambda ,\mu )fuzzy prime. □
Theorem 15 A (\lambda ,\mu )fuzzy ideal A of R is (\lambda ,\mu )fuzzy semiprime if and only if for all x\in R,
Proof The proof is similar to that of Theorem 14. □
Theorem 16 A (\lambda ,\mu )fuzzy ideal A of R is (\lambda ,\mu )fuzzy primary if and only if for all x,y\in R, \mathrm{\exists}{m}_{0}\in N such that
Proof Let A be (\lambda ,\mu )fuzzy primary. If possible, there exist {x}_{0},{y}_{0}\in R such that \lambda \vee A({x}_{0})\vee A({y}_{0}^{m})<A({x}_{0}{y}_{0})\wedge \mu for all m\in N, then \lambda \vee A({x}_{0})\vee A({y}_{0}^{m})<t\u2a7d\mu and {({x}_{0}{y}_{0})}_{t}\in A, where t=A({x}_{0}{y}_{0})\wedge \mu. So, \lambda \vee A({x}_{0})<t=t\wedge \mu and \lambda \vee A({y}_{0}^{m})<t=t\wedge \mu. From \lambda \vee A({x}_{0})<t, we have A({x}_{0})\u2a7d\lambda \vee A({x}_{0})<t\u2a7d\mu \u2a7d2\mu t and hence {x}_{0}\notin {A}_{t}^{(\lambda ,\mu )}. Similarly, {y}_{0}^{m}\notin {A}_{t}^{(\lambda ,\mu )}. This is a contradiction. Therefore, for all x,y\in R, \mathrm{\exists}{m}_{0}\in N such that \lambda \vee A(x)\vee A({y}^{{m}_{0}})\u2a7eA(xy)\wedge \mu.
Conversely, if for all x,y\in R, \mathrm{\exists}{m}_{0}\in N such that \lambda \vee A(x)\vee A({y}^{{m}_{0}})\u2a7eA(xy)\wedge \mu, then from {(xy)}_{t}\in A, we have \lambda \vee A(x)\vee A({y}^{{m}_{0}})\u2a7et\wedge \mu. So, \lambda \vee A(x)\u2a7et\wedge \mu or \lambda \vee A({y}^{{m}_{0}})\u2a7et\wedge \mu. That is, {x}_{t}{\in}_{(\lambda ,\mu )}A or {y}_{t}^{{m}_{0}}{\in}_{(\lambda ,\mu )}A. Hence, A is (\lambda ,\mu )fuzzy primary. □
Theorem 17 A (\lambda ,\mu )fuzzy ideal A of R is (\lambda ,\mu )fuzzy semiprimary if and only if for all x,y\in R, \mathrm{\exists}{m}_{0},{n}_{0}\in N such that
Proof The proof is similar to that of Theorem 16. □
Now, we characterize the (\lambda ,\mu )fuzzy prime (semiprimary) ideal by using its cut set.
Theorem 18 A fuzzy subset A of R is a (\lambda ,\mu )fuzzy prime (fuzzy semiprime) ideal if and only if for all t\in (\lambda ,\mu ], {A}_{t} is a prime (semiprime) ideal of R or {A}_{t}=\mathrm{\varnothing}.
Proof We only prove the case of a (\lambda ,\mu )fuzzy prime ideal.
Let A be a (\lambda ,\mu )fuzzy prime ideal of R. Then A is a (\lambda ,\mu )fuzzy ideal of R. So, {A}_{t} is an ideal of R or {A}_{t}=\mathrm{\varnothing} from Theorem 6. For all t\in (\lambda ,\mu ], if xy\in {A}_{t}, then \lambda \vee A(x)\vee A(y)\u2a7eA(xy)\wedge \mu \u2a7et\wedge \mu =t. Considering \lambda <t, we have A(x)\vee A(y)\u2a7et. It follows that x\in {A}_{t} or y\in {A}_{t}. Hence, {A}_{t} is a prime ideal of R.
Conversely, assume {A}_{t} is a prime ideal of R for all t\in (\lambda ,\mu ] whenever {A}_{t}\ne \mathrm{\varnothing}, then {A}_{t} is an ideal of R, and hence A is a (\lambda ,\mu )fuzzy ideal from Theorem 6. Let t\in (0,1] and {(xy)}_{t}\in A. If t\u2a7d\lambda, then it is clear that {x}_{t}{\in}_{(\lambda ,\mu )}A. If t\in (\lambda ,\mu ], then x\in {A}_{t} or y\in {A}_{t} since {A}_{t} is a prime ideal of R. So, {x}_{t}{\in}_{(\lambda ,\mu )}A or {y}_{t}{\in}_{(\lambda ,\mu )}A. If t>\mu, then xy\in {A}_{\mu}. It implies that x\in {A}_{\mu} or y\in {A}_{\mu}, since {A}_{\mu} is a prime ideal of R. Furthermore, we have
Similarly,
It follows that A is a (\lambda ,\mu )fuzzy prime ideal of R. □
Theorem 19 Let A be a (\lambda ,\mu )fuzzy ideal of R such that {A}_{\mu}\ne \mathrm{\varnothing}, and let B be a (\lambda ,\mu )fuzzy prime ideal of {A}_{\mu}. Then A\cap B is a (\lambda ,\mu )fuzzy prime ideal of {A}_{\mu}.
Proof From Theorem 1 and Theorem 6, {A}_{\mu} is a subring of R and A\cap B is a (\lambda ,\mu )fuzzy ideal of {A}_{\mu}. For all x,y\in {A}_{\mu}, t\in (0,1], we have A(x)\u2a7e\mu and A(y)\u2a7e\mu. Hence, x,y\in {A}_{t}^{(\lambda ,\mu )}. If {(xy)}_{t}\in A\cap B, then x\in {B}_{t}^{(\lambda ,\mu )} or y\in {B}_{t}^{(\lambda ,\mu )}, since B is a (\lambda ,\mu )fuzzy prime ideal of {A}_{\mu}. So x\in {A}_{t}^{(\lambda ,\mu )}\cap {B}_{t}^{(\lambda ,\mu )}={(A\cap B)}_{t}^{(\lambda ,\mu )}, or y\in {A}_{t}^{(\lambda ,\mu )}\cap {B}_{t}^{(\lambda ,\mu )}={(A\cap B)}_{t}^{(\lambda ,\mu )}. It follows that A\cap B is a (\lambda ,\mu )fuzzy prime ideal of {A}_{\mu}. □
Similarly, we have the following theorem.
Theorem 20 Let A be a (\lambda ,\mu )fuzzy ideal of R such that {A}_{\mu}\ne \mathrm{\varnothing}, and let B be a (\lambda ,\mu )fuzzy semiprime (fuzzy primary, fuzzy semiprimary) ideal of {A}_{\mu}. Then A\cap B is a (\lambda ,\mu )fuzzy semiprime (fuzzy primary, fuzzy semiprimary) ideal of {A}_{\mu}.
The following theorem gives the relation between a (\lambda ,\mu )fuzzy prime ideal with its preimage under a ring homomorphism.
Lemma 2 Let f:R\u27f6{R}^{\mathrm{\prime}} be a homomorphism of rings, and let B be a (\lambda ,\mu )fuzzy subring of {R}^{\mathrm{\prime}}. Then for all t\in (0,1], {f}^{1}{(B)}_{t}^{(\lambda ,\mu )}={f}^{1}({B}_{t}^{(\lambda ,\mu )}).
Theorem 21 Let f:R\u27f6{R}^{\mathrm{\prime}} be a homomorphism of rings, and let B be a (\lambda ,\mu )fuzzy prime ideal of {R}^{\mathrm{\prime}}. Then {f}^{1}(B) is a (\lambda ,\mu )fuzzy prime ideal of R.
Proof From Theorem 9, {f}^{1}(B) is a (\lambda ,\mu )fuzzy ideal of R. Let x,y\in R and t\in (0,1]. If {(xy)}_{t}\in {f}^{1}(B), then {(f(x)f(y))}_{t}\in B. Considering B is a (\lambda ,\mu )fuzzy prime ideal of {R}^{\mathrm{\prime}}, we have f(x)\in {B}_{t}^{(\lambda ,\mu )} or f(y)\in {B}_{t}^{(\lambda ,\mu )}. Hence x\in {f}^{1}{(B)}_{t}^{(\lambda ,\mu )} or y\in {f}^{1}{(B)}_{t}^{(\lambda ,\mu )} from Lemma 2. It follows that {f}^{1}(B) is a (\lambda ,\mu )fuzzy prime ideal of R. □
Similarly, we can obtain corresponding conclusions about a (\lambda ,\mu )fuzzy semiprime ideal, a (\lambda ,\mu )fuzzy primary ideal and a (\lambda ,\mu )fuzzy semiprimary ideal. But in general, the homomorphism image f(A) of a (\lambda ,\mu )fuzzy prime ideal A of R may not be (\lambda ,\mu )fuzzy prime even if f is a surjective homomorphism.
5 Conclusion
In this paper, we proposed the concept of a (\lambda ,\mu )fuzzy ideal which can be regarded as the generalization of a common fuzzy ideal introduced by Liu [11]. In the meantime, we also proposed several concepts of various (\lambda ,\mu )fuzzy ideals such as a (\lambda ,\mu )fuzzy prime ideal and a (\lambda ,\mu )fuzzy primary ideal, and then we characterized their properties and obtained several equivalence conditions of a (\lambda ,\mu )fuzzy prime ideal and a (\lambda ,\mu )fuzzy primary ideal.
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This work was supported by the National Natural Science Foundation of China (No. 71140008).
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Yao, B. Fuzzy prime ideals redefined. J Inequal Appl 2012, 203 (2012). https://doi.org/10.1186/1029242X2012203
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DOI: https://doi.org/10.1186/1029242X2012203
Keywords
 (\lambda ,\mu )fuzzy subring
 (\lambda ,\mu )fuzzy ideal
 (\lambda ,\mu )fuzzy prime ideal
 (\lambda ,\mu )fuzzy primary ideal