EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6153 ISSN 1307-5543 – ejpam.com Published by New York Business Global 1 A Study on Essential Fuzzy Ideals of Semirings2 Ronnason Chinram1, Saranya Hangsawat2,∗3 1 Division of Computational Science, Faculty of Science, Prince of Songkla University,4 Hat Yai, Songkhla 90110, Thailand5 2 Mathematics Program, Faculty of Science and Technology, Songkhla Rajabhat University,6 Songkhla 90000, Thailand7 8 Abstract. In this paper, we define essential fuzzy ideals of semirings and investigate some prop- erties of them. We give some properties of essential ideals and essential fuzzy ideals. Moreover, we show relationships between essential ideals and essential fuzzy ideals. 2020 Mathematics Subject Classifications: 16Y60, 03E729 Key Words and Phrases: Essential ideals, essential fuzzy ideals, minimal, prime, semiprime10 11 1. Introduction12 A semiring as the algebraic structure, is definitely a one of generalizations of rings. A13 semiring was appropriate to ask which properties of rings can be extended to semirings.14 The concept of semirings was introduced by Vandiver in 1935. One may expect semirings15 always to be extended to rings, however, Vandiver [1] gave examples of semirings that16 cannot be embedded in rings. Ideal theory is the main of researching of ring theory17 and also in semiring theory. A proper ideal of a ring is called essential if it has nonzero18 intersection with each nonzero ideal. Some properties of essential ideals of rings can see19 in [2]. Similar to ring theory, an essential ideal of a semiring was similar defined in [3].20 The notion of fuzzy sets as the extension of classical sets was introduced by Zadeh [4] in21 1965. Fuzzy sets permitted the gradual assessment of the membership of elements and22 described with the membership valued function of for each element in set to the value in23 the unit closed interval [0, 1]. Fuzzy sets have been applied to many algebraic structures24 like groups, semigroups, rings, modules and so on. Similar to many algebraic structures,25 fuzzy semirings were studied in [5]. The concepts of essential ideals were defined and26 studied in semigroups [6]. Moreover, their various kinds of fuzzifications of essential ideals27 of semigroups were also defined and studied [6–9]. Fuzzy essential ideals in rings were also28 studied in [10].29 ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6153 Email addresses: ronnason.c@psu.ac.th (R. Chinram), saranya.nu@skru.ac.th (S. Hangsawat) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Chinram, S. Hangsawat / Eur. J. Pure Appl. Math, 18 (3) (2025), 6153 2 of 7 The purpose of this paper is to define essential fuzzy ideals of semirings. Moreover, we30 show some relationships between essential ideals and their fuzzifications.31 2. Preliminaries32 In this section, we will recall in the basic concepts of semirings, fuzzy sets and fuzzy33 ideals of semirings.34 2.1. Semirings35 By a semiring we shall mean a nonempty set R endowed with two binary operations36 called the addition + and multiplication · satisfying the following conditions.37 (1) (R,+) is a commutative semigroup.38 (2) (R, ·) is a semigroup.39 (3) The multiplication distributes over the addition both from the left and from the40 right, that is, (a+ b)c = ac+ bc and c(a+ b) = ca+ cb for all a, b, c ∈ R.41 A semiring R is called commutative if ab = ba for all a, b ∈ R. An element 0 of a semiring42 R is called a zero of S if 0 + x = x and x0 = 0x = 0 for all x ∈ R. A semiring which43 contains a zero is called a semiring with zero. A non-empty subset I of a semiring R is44 called a left (resp. right ) ideal of R if for x, y ∈ I and r ∈ R imply that x + y ∈ I and45 rx ∈ I (resp. xr ∈ I). If I is both a left and right ideal of R, we say I is a two-sided ideal,46 or simply, an ideal of R. For x ∈ R, we let ⟨x⟩ be the smallest ideal of R generated by x.47 We have that ⟨x⟩ = {nx+ ax+ xb | n ∈ N0 and a, b ∈ R}.48 Let R be a commutative semiring with zero 0. Let x ∈ R \ {0}. If xy = 0 for some49 y ∈ R \ {0}, then x is called a zero divisor of R.50 2.2. Fuzzy Sets51 A fuzzy subset of a set S is a function from S into the closed interval [0, 1]. Let f and52 g be any two fuzzy subsets of a set S.53 1. The intersection of f and g is a fuzzy subset f ∩ g of S defined for all x ∈ S by54 (f ∩ g)(x) = min{f(x), g(x)}. 2. The union of f and g is a fuzzy subset f ∪ g of S defined for all x ∈ S by55 (f ∪ g)(x) = max{f(x), g(x)}. 3. If f(x) ≤ g(x) for all x ∈ S, we say that f is a subset of g and use the notation56 f ⊆ g.57 R. Chinram, S. Hangsawat / Eur. J. Pure Appl. Math, 18 (3) (2025), 6153 3 of 7 The support of a fuzzy subset f of a set S is defined by supp(f) = {x ∈ S | f(x) ̸= 0}.58 The characteristic mapping of a subset A of a set S is a fuzzy subset of S defined by59 CA(x) = { 1 if x ∈ A, 0 if x /∈ A. For any two subsets A and B of S, we have that CA∩B = CA ∩CB and CA∪B = CA ∪CB.60 A fuzzy set f of a semiring R is called a fuzzy ideal of R if for all x, y ∈ R, we have61 (1) f(x+ y) ≥ min{f(x), f(y)},62 (2) f(xy) ≥ max{f(x), f(y)}.63 Proposition 1. A nonempty subset A of a semiring R is an ideal of S if and only if CA64 is a fuzzy ideal of R.65 Proposition 2. Let f be a nonzero fuzzy ideal of a semiring R. Then supp(f) is an ideal66 of R.67 3. Main Results68 Throughout of this section, we let R be a semiring with zero 0. First, we recall the69 definition of essential ideals of R as follows:70 Definition 1. [3] An ideal I of R is called an essential ideal of R if I ∩K ̸= {0} for every71 nonzero ideal K of R.72 Example 1. We consider the semiring Z6 under the usual addition and multiplication of73 integers modulo 6. Let I = ⟨2⟩ = {0, 2, 4} and J = ⟨3⟩ = {0, 3}. We see that I ∩ J = {0}.74 Hence I and J are not essential ideals of Z6.75 Proposition 3. If R is commutative and I is an ideal containing a non zero divisor of76 R, then I is an essential ideal of R.77 Proof. Suppose that I is not an essential ideal of R. Then there exists a nonzero ideal78 K of R such that I ∩K = {0}. Let x is be a non zero divisor of I and y ∈ K \ {0}. Then79 xy ̸= 0 and xy ∈ I ∩K, this is a contradiction. Then I is an essential ideal of R.80 Example 2. We have that the semiring N0 := N ∪ {0} under the usual addition and81 multiplication of integers has no a zero divisor. Hence, by Proposition 3, every nonzero82 ideal of N0 is essential.83 The following corollary follows from Proposition 3.84 Corollary 1. If R is commutative and I is not an essential ideal of R, then every nonzero85 element in I is a zero divisor of R.86 R. Chinram, S. Hangsawat / Eur. J. Pure Appl. Math, 18 (3) (2025), 6153 4 of 7 However, the converse of Corollary 1 is not generally true.87 Example 3. We consider the semiring Z4 under the usual addition and multiplication of88 integers modulo 4. Let I = ⟨2⟩ = {0, 2}. We see that I is an essential ideal of Z4 and89 every nonzero element in I is a zero divisor of R because 2 · 2 = 0.90 The two following propositions are some properties of essential ideals of R.91 Proposition 4. Let I be an essential ideal of R. If J is an ideal of R containing I, then92 J is also an essential ideal of R.93 Proof. Assume that I is an essential ideal of R and J is an ideal of R such that I ⊆ J .94 Let K be any nonzero ideal of R. Thus I ∩K ̸= {0}. Since I ∩K ⊆ J ∩K, this implies95 that J ∩K ̸= {0}. Hence, J is an essential ideal of R.96 Proposition 5. Assume that R is commutative. Let I and J be essential ideals of R. If97 R has no a zero divisor, then I ∩ J is also an essential ideal of R.98 Proof. Since I and J are ideals of R, we have I ∩ J is also an ideal of R. Let K be99 any nonzero ideal of R. Thus I ∩K ̸= {0}. So there exists a nonzero element x ∈ I ∩K.100 Let y ∈ J ∖ {0}. Then xy ∈ (I ∩ J) ∩ K. By assumption, we have xy ̸= 0. Thus101 (I ∩ J) ∩K ̸= {0}. Hence, I ∩ J is an essential ideal of R.102 Definition 2. A fuzzy ideal f of R is called a nontrivial fuzzy ideal of R if there exists a103 nonzero element x ∈ R such that f(x) ̸= 0.104 By definition of nontrivial fuzzy ideals, the following lemma is obvious.105 Lemma 1. Let g be a fuzzy ideal of R. Then g is a nontrivial fuzzy ideal of R if and only106 if supp(g) ̸= {0}.107 We define the definition of essential fuzzy ideals of R as follows:108 Definition 3. A fuzzy ideal f ofR is called an essential fuzzy ideal ofR if supp(f∩g) ̸= {0}109 for every nontrivial fuzzy ideal g of S.110 Example 4. We consider the semiring N0 under the usual addition and multiplication of111 integers. Define a fuzzy subset f of N0 by f(0) = 1 and f(n) = n− 1 n for all n ∈ N. We112 have that f is an essential fuzzy ideal of N0.113 Proposition 6. Let f be an essential fuzzy ideal of R. If h is a fuzzy ideal of R such that114 f ⊆ h, then h is also essential.115 Proof. Assume that f is an essential fuzzy ideal of R and let h be a fuzzy ideal of S116 such that f ⊆ h. Let g be any nontrivial fuzzy ideal of R. Thus supp(f ∩ g) ̸= {0}. So117 supp(h ∩ g) ̸= {0}. Hence, h is also an essential fuzzy ideal of R.118 The two following theorems show relationships between essential ideals and essential119 fuzzy ideals of R.120 R. Chinram, S. Hangsawat / Eur. J. Pure Appl. Math, 18 (3) (2025), 6153 5 of 7 Theorem 1. A nonzero ideal I of R is essential if and only if CI is an essential fuzzy121 ideal of R.122 Proof. Assume that I is an essential ideal of R. By Proposition 1, CI is a fuzzy ideal123 of R. We let g be any nontrivial fuzzy ideal of R. By Lemma 1 and Proposition 2, we have124 supp(g) is a nonzero ideal of S. So I∩supp(g) ̸= {0}. Then there exists a nonzero element125 x such that x ∈ I ∩supp(g), this implies that (CI ∩g)(x) ̸= 0. Therefore x ∈ supp(CI ∩g).126 Thus supp(CI ∩g) ̸= {0}. Hence CI is an essential fuzzy ideal of R. To prove the converse,127 we assume that CI is an essential fuzzy ideal of R. Let K be a nonzero ideal of R. By128 Lemma 1, we have that CK is a nontrivial fuzzy ideal of R. Thus supp(CI ∩ CK) ̸= {0}.129 Thus CI∩K ̸= C{0}, this implies that I ∩K ̸= {0}. Hence I is essential.130 Theorem 2. A nontrivial fuzzy ideal f of R is essential if and only if supp(f) is an131 essential ideal of R.132 Proof. Assume that f is an fuzzy essential ideal of R. Since f is a fuzzy ideal of R,133 supp(f) is an ideal of R by Proposition 2. Let J be any nonzero ideal of R. By Lemma 1,134 CJ is a nontrivial fuzzy ideal of R. Since f is essential, supp(f ∩ CJ) ̸= {0}. Thus there135 exists a nonzero element x in R such that (f ∩CJ)(x) ̸= 0. Thus f(x) ̸= 0 and CJ(x) ̸= 0.136 Hence x ∈ supp(f)∩ J . Then supp(f)∩ J ̸= {0}, this implies that supp(f) is an essential137 ideal of R. Conversely, assume that supp(f) is an essential ideal of R. Let g be a nontrivial138 fuzzy ideal of R. Thus supp(g) is a nonzero ideal of R. So supp(f)∩ supp(g) ̸= {0}. Thus139 there exists a nonzero element x in R such that x ∈ supp(f) ∩ supp(g), so f(x) ̸= 0 and140 g(x) ̸= 0. Therefore, (f ∩ g)(x) ̸= 0. Hence, supp(f ∩ g) ̸= {0}. This implies that f is141 essential.142 In the remainder of this section, we will investigate relationships between (minimal,143 prime, semiprime) essential ideals and (minimal, prime, semiprime) essential fuzzy ideals.144 An essential ideal I of R is called minimal if for every essential ideal J of R such145 that J ⊆ I, we have J = I. An essential fuzzy ideal f of R is called minimal if for every146 essential fuzzy ideal g of R such that g ⊆ f , we have supp(f) = supp(g).147 Theorem 3. A nonempty subset S of R is a minimal essential ideal of R if and only if148 CS is a minimal essential fuzzy ideal of R.149 Proof. Assume that S is a minimal essential ideal of R. Since S is an essential ideal150 of R, we have CS is an essential fuzzy ideal of R by Theorem 1. Let g be any essential151 fuzzy ideal of R such that g ⊆ CS . Thus supp(g) ⊆ supp(CS) = S. Since g is an essential152 fuzzy ideal of R, by Theorem 2, supp(g) is an essential ideal of R. Therefore supp(g) = S153 because S is minimal. This implies that supp(g) = supp(CS). Hence, CS is minimal.154 To prove the converse, we suppose that CS is a minimal essential fuzzy ideal of R155 and let I be an essential fuzzy ideal of R such that I ⊆ S. This implies that CI is an156 essential fuzzy ideal of R such that CI ⊆ CS . Since CS is minimal, supp(CI) = supp(CS).157 Therefore, I = supp(CI) = supp(CS) = S. Hence, S is minimal.158 An essential ideal I of R is called prime if rs ∈ I implies r ∈ I or s ∈ I for all r, s ∈ R.159 An essential fuzzy ideal f of R is called prime if f(rs) ≤ max{f(r), f(s)} for all r, s ∈ R.160 R. Chinram, S. Hangsawat / Eur. J. Pure Appl. Math, 18 (3) (2025), 6153 6 of 7 Theorem 4. A nonempty subset S of R is a prime essential ideal of R if and only if CS161 is a prime essential fuzzy ideal of R.162 Proof. Let S be a prime essential ideal of R. By Theorem 1, CS is an essential fuzzy163 ideal of R. Let r and s be any two elements of R. If rs ∈ S, then we have that r ∈ S or164 s ∈ S because S is prime. Thus max{CS(r), CS(s)} = 1 ≥ CS(rs). Otherwise, if rs ̸∈ S,165 then CS(rs) = 0 ≤ max{CS(r), CS(s)}. By both two cases, we conclude that CS is a166 prime essential fuzzy ideal of R. Conversely, we assume that CS is a prime essential fuzzy167 ideal of R. It follows by Theorem 1 that S is an essential ideal of R. Let r and s be168 two elements of R such that rs ∈ S. This implies that CS(rs) = 1. Since CS is prime,169 CS(rs) ≤ max{CS(r), CS(s)}. Then max{CS(r), CS(s)} must be equal to 1 and so r ∈ S170 or s ∈ S. Hence, S is a prime essential ideal of R.171 An essential ideal I of R is called semiprime if for all r ∈ R, r2 ∈ I implies r ∈ I. An172 essential fuzzy ideal f of R is called semiprime if for all r ∈ R, f(r2) ≤ f(r).173 Theorem 5. A nonempty subset S of R is a semiprime essential ideal of R if and only174 if CS is a semiprime essential fuzzy ideal of R.175 Proof. Let S be a semiprime essential ideal of R. By Theorem 1, CS is an essential176 fuzzy ideal of R. Let r be any element in R. If r2 ∈ S, then we have r ∈ S because S is177 semiprime. This implies that CS(r) = 1. Hence, CS(r) ≥ CS(r 2). Otherwise, if r2 ̸∈ S,178 then CS(r 2) = 0 ≤ CS(r). By both two cases, we conclude that CS is a semiprime essential179 fuzzy ideal of R. To prove the converse, assume that CS is a semiprime essential fuzzy180 ideal of R. By Theorem 1, we have that S is an essential ideal of R. Let r ∈ R such that181 r2 ∈ S. So CS(r 2) = 1. Since CS is semiprime, CS(r 2) ≤ CS(r). Since CS(r 2) = 1, CS(r)182 must be equal 1. Therefore, r ∈ S. Hence, S is a semiprime essential ideal of R.183 4. Conclusion184 Let R be a semiring with zero. In this paper, we first show some properties of essential185 ideals of R (Proposition 3-5 and Corollary 1). Next, we introduce the definition of essential186 fuzzy ideals of R (Definition 3). Finally, we show relationships between essential ideals187 and essential fuzzy ideals of R in Theorem 1-5.188 In the future work, we can define an essential ideal of other algebraic structures and189 show their properties. 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