Fuzzy prime ideals redefined

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 ‘quasi-coincident 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 $A_t=\{x\in G\mid A(x)⩾t\}$ 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 $A_{〈t〉}$ of A are defined as follows:

$$A_t=\{x\in X\mid A(x)\ge t\},A_{〈t〉}=\{x\in X\mid A(x)>t\}.$$

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

$$A(y)=\{\begin{array}{cc}t,\hfill & \text{if }y=x,\hfill \\ 0,\hfill & \text{if }y\ne x\hfill \end{array}$$

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 quasi-coincident with) a fuzzy subset A, written as $x_t\in A$ (resp. $x_{t}qA$), if

$$A(x)\ge t(\text{resp. }A(x)+t>1),$$

$x_t\in A$ 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) $x_t,y_r\in A⟹{(x+y)}_{t\wedge r}\in \vee qA$,

• (2) $x_t\in A⟹{(-x)}_t\in \vee qA$,

• (3) $x_t,y_r\in A⟹{(xy)}_{t\wedge r}\in \vee qA$.

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) $x_t\in A,y\in R⟹{(xy)}_t,{(yx)}_t\in \vee qA,\mathrm{\forall }t\in (0,1]$.

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) $A(x-y)\ge A(x)\wedge A(y)\wedge 0.5$,

• (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 $x,y\in R$,

$$A(x+y)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu ,A(-x)\vee \lambda \ge A(x)\wedge \mu .$$

Clearly, a fuzzy subset A of R is a $(\lambda ,\mu )$-fuzzy addition subgroup of R if and only if for all $x,y\in R$, $A(x-y)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu$.

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) $A(x-y)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu$,

• (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) $A(x-y)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu$,

• (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) $A(x-y)\vee \lambda \ge A(x)\wedge A(y)\wedge \mu$,

• (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

$$(A+B)(x)=sup\{A(x_1)\wedge B(x_2)\mid x=x_1+x_2\},\mathrm{\forall }x\in R.$$

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: $\mathrm{\forall }x\in R$,

$$(A\odot B)(x)=\{\begin{array}{c}sup\{{inf}_{1⩽i⩽n}(A(x_i)\wedge B(y_i))\mid x={\sum }_{i=1}^n{x}_i{y}_i,x_i,y_i\in R,n\in N\},\hfill \\ \phantom{0,}\text{if }x\text{ can be expressed as }x=\sum x_i{y}_i,x_i,y_i\in R,\hfill \\ 0,\text{otherwise}.\hfill \end{array}$$

 □

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 $t\in [\lambda ,\mu )$, $A_{〈t〉}$ is a subring (ideal) of R or $A_{〈t〉}=\mathrm{\varnothing }$.

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 $x,y\in A_{〈t〉}$, then $A(x)\wedge A(y)>t$, $A(x-y)\vee \lambda ⩾A(x)\wedge A(y)\wedge \mu >t$. Considering $\lambda ⩽t$, we have $A(x-y)>t$ and $x-y\in A_{〈t〉}$. Similarly, $xy\in A_{〈t〉}$. It follows that $A_{〈t〉}$ is a subring of R.

Conversely, assume that $A_{〈t〉}$ 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 $A(x_0-y_0)⩽t$, $A(x_0)\wedge A(y_0)>t$. So, $x_0,y_0\in A_{〈t〉}$, and $x_0-y_0\notin A_{〈t〉}$. This is a contradiction to the fact that $A_{〈t〉}$ is a subring of R. This shows that $A(x-y)\vee \lambda ⩾A(x)\wedge A(y)\wedge \mu$ holds for all $x,y\in R$.

Similarly, $A(xy)\vee \lambda ⩾A(x)\wedge A(y)\wedge \mu$, $\mathrm{\forall }x,y\in R$. That is, $A_{〈t〉}$ 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

$$f(A)(y)=\{\begin{array}{cc}sup\{A(x)\mid f(x)=y\},\hfill & \mathit{\text{if}}{f}^{-1}(y)\ne \mathrm{\varnothing },\hfill \\ 0,\hfill & \mathit{\text{otherwise}}\hfill \end{array}\mathrm{\forall }y\in R^{\mathrm{\prime }}.$$

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

$$f^{-1}(B)(x)=B(f(x)),\mathrm{\forall }x\in R.$$

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

$$A_t^{(\lambda ,\mu )}=\{x\in X\mid A(x)\vee \lambda ⩾t\wedge \mu \text{ or }A(x)>(2\mu -t)\vee \lambda \}$$

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

$$A_t^{(\lambda ,\mu )}=\{\begin{array}{cc}X,\hfill & t⩽\lambda ,\hfill \\ A_t,\hfill & \lambda <t⩽\mu ,\hfill \\ A_{〈(2\mu -t)\vee \lambda 〉},\hfill & t>\mu .\hfill \end{array}$$

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 $t\in [0,1]$,

$$A\subseteq B⟹A_t^{(\lambda ,\mu )}\subseteq B_t^{(\lambda ,\mu )}.$$

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 $A(x)\vee \lambda ⩾t\wedge \mu$ and $B(x)\vee \lambda ⩾t\wedge \mu$, then $(A\cap B)(x)\vee \lambda ⩾t\wedge \mu$. 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 $A(x)\vee \lambda ⩾t\wedge \mu$ and $B(x)>(2\mu -t)\vee \lambda$, then when $t⩽\mu$, we have $B(x)\vee \lambda >(2\mu -t)\vee \lambda ⩾\mu ⩾t\wedge \mu$ and hence $(A\cap B)(x)\vee \lambda ⩾t\wedge \mu$. When $t>\mu$, we have $A(x)⩾\mu >(2\mu -t)\vee \lambda$ 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 $B(x)\vee \lambda ⩾t\wedge \mu$, 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 $A(x)\vee B(x)\vee \lambda ⩾t\wedge \mu$ or $A(x)\vee B(x)>(2\mu -t)\vee \lambda$. It means that $A(x)\vee \lambda ⩾t\wedge \mu$, or $B(x)\vee \lambda ⩾t\wedge \mu$, 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 $t⩽\lambda$, then $A_t^{(\lambda ,\mu )}=R$. If $\lambda <t⩽\mu$, then $A_t^{(\lambda ,\mu )}=A_t$ and $A_t$ is a subring of R from Theorem 6. If $t>\mu$, then $A_t^{(\lambda ,\mu )}=A_{〈(2\mu -t)\vee \lambda 〉}$ 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 $ab\in S⟹a\in S$ 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 $t\in (0,1]$, ${(x^2)}_t\in A⟹x_t\in \vee qA$,

• (2) fuzzy prime if for all $x,y\in R$ and $t\in (0,1]$, ${(xy)}_t\in A⟹x_t\in \vee qA$ or $y_t\in \vee qA$,

• (3) fuzzy semiprimary if for all $x,y\in R$ and $t\in (0,1]$, ${(xy)}_t\in A⟹x_t^m\in \vee qA$ or $y_t^n\in \vee qA$ for some $m,n\in N$,

• (4) fuzzy primary if for all $x,y\in R$ and $t\in (0,1]$, ${(xy)}_t\in A⟹x_t\in \vee qA$ 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 $t\in (0,1]$, ${(x^2)}_t\in A⟹x_t{\in }_{(\lambda ,\mu )}A$,

• (2) $(\lambda ,\mu )$-fuzzy prime if for all $x,y\in R$ and $t\in (0,1]$, ${(xy)}_t\in A⟹x_t{\in }_{(\lambda ,\mu )}A$ or $y_t{\in }_{(\lambda ,\mu )}A$,

• (3) $(\lambda ,\mu )$-fuzzy semiprimary if for all $x,y\in R$ and $t\in (0,1]$, ${(xy)}_t\in A⟹x_t^m{\in }_{(\lambda ,\mu )}A$ 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 $t\in (0,1]$, ${(xy)}_t\in A⟹x_t{\in }_{(\lambda ,\mu )}A$ 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 $x,y\in R$,

$$\lambda \vee A(x)\vee A(y)⩾A(xy)\wedge \mu .$$

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 $A(x_0)⩽\lambda \vee A(x_0)<t⩽\mu ⩽2\mu -t$ 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 $\lambda \vee A(x)\vee A(y)⩾A(xy)\wedge \mu$.

Conversely, if for all $x,y\in R$, $\lambda \vee A(x)\vee A(y)⩾A(xy)\wedge \mu$ and ${(xy)}_t\in A$, then $\lambda \vee A(x)\vee A(y)⩾t\wedge \mu$. So, $\lambda \vee A(x)⩾t\wedge \mu$ or $\lambda \vee A(y)⩾t\wedge \mu$. 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 $x\in R$,

$$\lambda \vee A(x)⩾A\left(x^2\right)\wedge \mu .$$

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

$$\lambda \vee A(x)\vee A\left(y^{m_0}\right)⩾A(xy)\wedge \mu .$$

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 $\lambda \vee A(x_0)\vee A(y_0^m)<t⩽\mu$ 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 $A(x_0)⩽\lambda \vee A(x_0)<t⩽\mu ⩽2\mu -t$ 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 $\lambda \vee A(x)\vee A(y^{m_0})⩾A(xy)\wedge \mu$.

Conversely, if for all $x,y\in R$, $\mathrm{\exists }{m}_0\in N$ such that $\lambda \vee A(x)\vee A(y^{m_0})⩾A(xy)\wedge \mu$, then from ${(xy)}_t\in A$, we have $\lambda \vee A(x)\vee A(y^{m_0})⩾t\wedge \mu$. So, $\lambda \vee A(x)⩾t\wedge \mu$ or $\lambda \vee A(y^{m_0})⩾t\wedge \mu$. 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

$$\lambda \vee A\left(x^{m_0}\right)\vee A\left(y^{n_0}\right)⩾A(xy)\wedge \mu .$$

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 $\lambda \vee A(x)\vee A(y)⩾A(xy)\wedge \mu ⩾t\wedge \mu =t$. Considering $\lambda <t$, we have $A(x)\vee A(y)⩾t$. 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 $t⩽\lambda$, 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

$$x\in A_{\mu }⟹A(x)⩾\mu =t\wedge \mu ⟹x\in A_t^{(\lambda ,\mu )}⟹x_t{\in }_{(\lambda ,\mu )}A.$$

Similarly,

$$y\in A_{\mu }⟹y_t{\in }_{(\lambda ,\mu )}A.$$

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 $A(x)⩾\mu$ and $A(y)⩾\mu$. 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 $f:R⟶R^{\mathrm{\prime }}$ 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 $f:R⟶R^{\mathrm{\prime }}$ 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.