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FUZZY INFORMATION AND ENGINEERING 2021, VOL. 13, NO. 2, 223–235 https://doi.org/10.1080/16168658.2021.1950390 On the Primary Decomposition of k-Ideals and Fuzzy k-Ideals in Semirings a a a b Ram Parkash Sharma , Madhu Dadhwal ,Richa Sharma and S. Kar Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla, India; Department of Mathematics, Jadavpur University, Kolkata, India ABSTRACT ARTICLE HISTORY Received 13 June 2020 Observing that every k-irreducible ideal of a semiring R is a k- Revised 13 May 2021 primary ideal, if R is an additively cancellative, yoked and commu- Accepted 9 June 2021 tative Noetherian semiring, we establish the primary decomposition and uniqueness of the primary decomposition of k-ideals of such KEYWORDS semirings. Finally, the primary decomposition and uniqueness of pri- Semirings; fuzzy k-ideals; mary decomposition proved for k-ideals is also generalised for fuzzy primary ideals; primary decomposition k-ideals of these semirings. AMS MATHEMATICS SUBJECT CLASSIFICATION (2020) 16Y60 1. Introduction Ideals play an important role in both ring theory and semiring theory. But in the absence of additive inverses in semirings, the structure of ideals in semirings differs from that of ring theory. The ideals in semirings possessing the very obvious property of ideals of rings, ‘if x + y ∈ I, x ∈ I, then y ∈ I’ are known as k-ideals and the role of these ideals in semirings becomes significant in the absence of additive inverses. The results which are true for ideals in rings have also been established for k-ideals in semirings by various authors (cf. [1–13]). In view of these facts various researchers attempted the primary decomposition for k-ideals in semirings analogous to the primary decomposition of ideals in rings: In a commutative Noetherian ring, every ideal can be decomposed as a finite intersection of primary ideals (Lasker–Noether Theorem [14]). For a deep study of primary ideals in rings and semirings one can refer to [8–12, 15–19]. The above result of ring theory is not true for arbitrary ideals in semirings as noticed in [20]. Atani and Atani [20, Theorem 4] had proved that in a commutative Noetherian semir- ing, every proper k-ideal can be represented as a finite intersection of k-primary ideals. But it was observed by Lescot [21] that there were some errors in the results used to prove the aforementioned result. For example, I + Ra is not a k-ideal, even if I is a k-ideal. But in [20], it is taken for granted that the ideal I + Ra is a k-ideal. Lescot [21] found these errors after observing in Example 6.2 that {0} ideal may not be a finite intersection of k-primary ideals in a commutative Noetherian semiring. With these observations, he developed CONTACT Madhu Dadhwal mpatial.math@gmail.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (http:// creativecommons.org/licenses/by-nc/4.0/), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. 224 R. P. SHARMA ET AL. the theory of weak primary decomposition for semirings of characteristic 1. But still the question of settling the primary decomposition for a proper k-ideal (other than {0} and semiring R) remained unsolved. In this direction, Kar et al. [22, Theorem 4.4] proved that every proper k-ideal of a commutative Noetherian semiring R canbeexpressedasafinite intersection of k-irreducible ideals (Definition 2.1). They tried to prove the primary decom- position by observing that in a commutative Noetherian semiring, every k-irreducible ideal is a k-primary ideal [22, Theorem 4.6]. But the proof of this result has the same errors again and the problem still remains unsolved. This motivated us to establish the primary decomposition of k-ideals in commutative Noetherian semirings. In this paper, we provide a correct proof of Theorem 4.6 of [22], which settles the pri- mary decomposition for k-ideals in commutative Noetherian semirings. In order to establish the uniqueness of primary decomposition for semirings, first we prove some required basic results for uniqueness and finally establish the uniqueness of primary decomposition for k-ideals in commutative Noetherian semirings: If I =∩ Q , where each Q is a primary i i i=1 ideal and Q = P , then the set {P , P , ... ., P } is independent of particular choice of i i 1 2 n decomposition of I. Fuzzy k-ideals play an important role for the study of different classes of semirings. These ideals have been studied by many authors (cf. [1, 7, 18, 23–27]). From the primary decompo- sition of k-ideals in semirings, the fuzzy primary decomposition of fuzzy k-ideals follows as given in [22]. But the uniqueness theorem for fuzzy k-ideals [22, Theorem 5.5], is also incor- rect. This motivated us to reinvestigate the uniqueness of fuzzy k-primary decomposition in semirings. 2. Primary Decomposition of k-Ideals in Semirings Throughout this paper, R is a commutative semiring with identity. First, we recall some definitions and results which are necessary to prove the primary decomposition and uniqueness theorems for k-ideals in commutative Noetherian semirings. Definition 2.1: Aproper k-ideal I of a semiring R is called k-irreducible ideal if for any two k-ideals J, K of R, I = J ∩ K implies that either I = J or I = K. Definition 2.2: Let I be a k-ideal of a semiring R. Then I is said to have a primary decomposi- tion if I canbeexpressedas I =∩ Q , where each Q is a primary ideal of R. Also, a primary i i i=1 decomposition of the type I =∩ Q with Q = P , is called a reduced primary decompo- i i i i=1 sition of I,if P ’s are distinct and I cannot be expressed as an intersection of a proper subset of ideals Q in the primary decomposition of I. A reduced primary decomposition can be obtained from any primary decomposition by deleting those Q that contains ∩ Q and j i i=1 i=j grouping together all distinct Q ’s-primary ideal. The main aim of this section is to prove the existence and uniqueness of primary decom- position for k-ideals in a commutative Noetherian semiring. First, we prove the existence part as stated below: Theorem 2.3 (Primary Decomposition of k-Ideals): Let I be a k-ideal of an additively can- cellative, yoked and commutative Noetherian semiring R. Then I can be represented as a finite intersection of primary ideals of R, that is, I has a reduced primary decomposition. FUZZY INFORMATION AND ENGINEERING 225 The decomposition of k-ideals in terms of k-irreducible ideals has already been proved in [22] as follows: Theorem 2.4 ([22], Theorem 4.4): Every proper k-ideal of a commutative Noetherian semir- ing R can be expressed as a finite intersection of k-irreducible ideals. Therefore, primary decomposition for k-ideals in semirings follows immediately if we prove Theorem 2.5: Let R be an additively cancellative, yoked and commutative Noetherian semir- ing. Then every k-irreducible ideal of R is a primary ideal of R. Kar et al. [22, Theorem 4.6], made an attempt to prove the same without considering additively cancellative and yoked semiring, but the proof of the result has some errors as mentioned in the introduction. For the proof of the required result, we first analyse some common errors committed by many authors regarding k-ideals, and prove some facts about k-ideals. First, we note that any ideal I =< a > is not a k-ideal in semirings as observed by Golan in [28, Example 6.17]. That is, if I = <(1 + x)> is an ideal of N[x] (semiring of 3 3 polynomials over non-negative integers N in the indeterminate x), then (1 + x) = (x + 1) + 3x(1 + x) ∈ I,3x(1 + x) ∈ I, but (x + 1)/ ∈ I implies that I is not a k-ideal of N[x]. Any ideal generated by a single element becomes a k-ideal, if we impose some conditions on a semiring as shown below: Lemma 2.6: Let R be an additively cancellative, yoked and zerosumfree semiring. Then for any a ∈ R, the ideal I =< a > isak-idealofR. Proof: Let a + a , a ∈ I. Then there exist some r , r ∈ R such that a + a = ar and 1 2 1 1 2 1 2 1 a = ar .As R is yoked, there exists some r ∈ R such that r + r = r or r + r = r .If r + 1 2 3 1 3 2 2 3 1 1 r = r , then ar = ar + a = ar + ar + a implies that ar + a = 0, as R is additively 3 2 1 2 2 1 3 2 3 2 cancellative. Also, a = ar = 0 ∈ I,as R is zerosumfree. 2 3 Further, if r + r = r , then ar + a = ar = a(r + r ) = ar + ar implies that a = 2 3 1 2 2 1 2 3 2 3 2 ar ∈ I, since R is additively cancellative. Thus, I =< a > is a k-ideal of R. The semiring considered in above example is additively cancellative, zerosumfree, but it is not yoked, for let 2 2 f (x) = 5x + 9x +2and g(x) = 11x + 3x + 5 be two polynomials in N[x], then there exists no h(x) ∈ N[x] such that either f (x) + h(x) = g(x) or g(x) + h(x) = f (x). Similar to an ideal generated by a single element, the sum of two k-ideals may not be a k-ideal in a semiring. There are plenty of k-ideals in the semiring N, but their sum is not a k-ideal. However, the sum of two k-ideals is a k-ideal in a lattice ordered semiring ( cf. [28, Corollary 21.22]). While proving Theorem 2.5, the authors wrongly used that the ideals (Q+ < a >) and (Q+ < b >) are k-ideals. In view of the above observations, Theorem 2.5 follows verbatim as proved in [22, Theorem 4.6] for additively cancellative, yoked, zerosum- free and lattice ordered semirings, because in this case, both an ideal generated by a single 226 R. P. SHARMA ET AL. element and sum of two k-ideals are k-ideals. Here, we give a proof of Theorem 2.5 without resorting to the restrictions: lattice ordered and zerosumfree semiring. Proof of Theorem 2.5.: Let Q be a k-irreducible ideal of a Noetherian semiring R.Let ab ∈ Q be such that b ∈ / Q. Now, we construct two ideals I and J of R as follows: I =< a > +Q and J =< b > +Q. Then, clearly, Q ⊆ I ∩ J. Let y ∈ I ∩ J. Then y = a z + q for some z ∈ R and q ∈ Q.Again aJ ⊆ Q ( since ab ∈ Q) and so ay ∈ Q ( since y ∈ J). n+1 n+1 Therefore, ay = a z + aq. Thus, we get that a z + aq ∈ Q. Also, aq ∈ Q, since q ∈ Q n+1 and Q is an ideal of R. It follows that a z ∈ Q, since Q is a k-ideal of R. Construct a set A = {x ∈ R | a x ∈ Q}. It is easy to check that A is an ideal of R and A ⊆ A ··· . is an ascending n 1 2 + n+1 chain of ideals. Since R is Noetherian, so A = A = ··· . for some n ∈ Z .Again a z ∈ n n+1 Q implies that z ∈ A = A . It demonstrates that a z ∈ Q which implies that y ∈ Q and n+1 n hence I ∩ J = Q. Let rad(R) denote the Jacobson Bourne radical of a semiring R, that is, the intersection of all maximal k-ideals of R,as R is assumed to be additively cancellative and yoked (cf. [29, Proposition 23]). Assume that A is an ideal of R and rad(A) denotes the intersection of all maximal k-ideals of R containing A. ¯ ¯ Let A denotes the k-closure of an ideal A of R, i.e.A ={a ∈ A | a + b = c, for some b, c ∈ A} and I ={x ∈ R | a ∈ I for some positive integer n}. Obviously, I is a k-ideal of R and every Noetherian semiring is weakly Noetherian (a semiring R is weakly Noetherian if every ascending chain of k-ideals of R is ultimately sta- tionary). Then by Lescot [21, Corollary 6.6], there are prime k-ideals P , P , ... ., P of R such 1 2 n √ √ that I = P ∩ P ··· . ∩ P . Thus, we have I ⊆ rad(R) and by Lescot [21, Lemma 2.2], 1 2 n √ √ √ I ∩ J ⊆ I ∩ J = I ∩ J ⊆ rad(I ∩ J). Therefore, rad(I ∩ J) ⊆ rad(I ∩ J). Again, I ∩ J ⊆ I ∩ J implies that rad(I ∩ J) ⊆ rad(I ∩ J).Thus, rad(I ∩ J) = rad(I ∩ J) = rad(I) ∩ rad(J).So, we have rad(I ∩ J) = rad(I) ∩ rad(J). Now, Q = I ∩ J which implies that Q = I ∩ J ⇒ Q = I ∩ J, since Q is a k-ideal of R.This shows that rad(Q) = rad(I ∩ J) = rad(I) ∩ rad(J), that is, rad(Q) = rad(I) ∩ rad(J). Again, Q is a k-irreducible ideal, which implies that rad(Q) is k- irreducible. Also rad(Q) = Q ∩ rad(R), but rad(Q) = rad(R).Thus, rad(Q) = Q, since rad(Q) is k-irreducible. Accord- ingly, Q = rad(I) ∩ rad(J), where each of rad(I) and rad(J) are k-ideals of R.Now b ∈ J implies that b ∈ rad(J), but b ∈ / Q, that is, Q = rad(J).So Q = rad(I), since Q is k-irreducible. n n Further, a ∈ I leads to a ∈ rad(I) = Q. Hence, Q is a primary ideal. Remark 2.7: Now Theorem 2.3, follows by combining Theorems 2.4 and 2.5. It is impor- tant to note here that all Q ’s in the primary decomposition of a k-ideal I have an additional property that these are also k-ideals. Now it only remains to prove the uniqueness of primary decomposition of a k-ideal I. The following lemma will be used to prove the uniqueness of reduced primary decom- position of k-ideals in semirings. FUZZY INFORMATION AND ENGINEERING 227 Lemma 2.8: Let R be a semiring and Q a P-primary ideal of R. Then we have (i) If x ∈ R, (Q : x) ={r ∈ R | rx ∈ Q} is an ideal of R; (ii) If x ∈ Q, then (Q : x) = R; (iii) If x ∈ / P, then (Q : x) = Q; (iv) If x ∈ / Q, then (Q : x) is P-primary. Proof: (i) Let r , r ∈ (Q : x) and a ∈ R. Then r x, r x ∈ Q implies that (r + r )x ∈ Q and 1 2 1 2 1 2 r ax ∈ Q,as Q is an ideal of R. Hence (Q : x) is an ideal of R. (ii) If x ∈ Q, then Rx ⊆ Q,as Q is an ideal of R which implies that R ⊆ (Q : x). Also, (Q : x) is an ideal of R and so, (Q : x) = R. (iii) Assume that x ∈ / P.If a ∈ Q, then ax ∈ Q implies that a ∈ (Q : x). For the converse part, suppose that b ∈ / Q and xb ∈ Q. Then x ∈ Q = P,as Q is P-primary ideal of R which contradicts that x ∈ / P. This infers that xb ∈ / Q and so b ∈ / (Q : x). (iv) Suppose that x ∈ / Q.If y ∈ (Q : x), then xy ∈ Q implies that y ∈ Q = P,as Q is P- √ √ √ √ primary ideal of R.Thus, Q ⊆ (Q : x) ⊆ P implies that P = Q ⊆ (Q : x) ⊆ P. Also, P = P,as P is a prime ideal of R and so P = (Q : x). Now, we show that (Q : x) is a primary ideal of R. Clearly, the ideal (Q : x) is a proper as x ∈ / Q and so 1 ∈ / (Q : x). Assume that ab ∈ (Q : x) and b ∈ / (Q : x) for a, b ∈ R. Then abx ∈ Q and Q is a P-primary ideal of R which implies that √ √ either ax ∈ Q or b ∈ P = (Q : x).Thus, a ∈ (Q : x) as b ∈ / (Q : x). We now prove the uniqueness of the reduced primary decomposition of a k-ideal of a semiring as follows: Theorem 2.9 (Uniqueness of Primary Decomposition): Let R be an additively cancellative, yoked and commutative Noetherian semiring and I be a k-ideal of R. If I =∩ Q is a reduced i=1 primary decomposition of I with Q = P for i = 1, 2 ··· .n, then i i {P , P , ... ., P }={Prime ideals P |there exists x ∈ R such that P = (I : x)}. 1 2 n The set {P ,P ··· .P } is independent of the particular reduced primary decomposition chosen 1 2 n for I. Proof: Let x ∈ R. Then by Lemma 2.8 (iv), n n (I : x) = ∩ Q : x = ∩ (Q : x)=∩ P i i i i=1 i=1 i, x∈ /Q √ √ and therefore, (I : x) ⊆ P , for all i = 1, 2 ··· .n. Also, if (I : x) is prime, then P ⊆ i i, x∈ /Q √ √ ∩ P = (I : x) implies that P ⊆ (I : x) for some i = 1, 2 ··· n. i,x∈ /Q i i Thus, we have {P , P , ... ., P }⊆{Prime ideals P |there exists x ∈ R such that P = (I : x)}. 1 2 n On the other hand, for i ∈{1, 2 ··· .n},wehave ∩ Q Q , as the primary decomposition j i j=i is reduced. So there exists some x ∈∩ Q and x ∈ / Q .If y ∈ (Q : x ), then yx ∈ Q and i j i i i i i i j=i yx ∈ (∩ Q ) ∩ Q = I which implies that y ∈ (I : x ). i j i i j=i 228 R. P. SHARMA ET AL. Thus, (Q : x ) ⊆ (I : x ) ⊆ (Q : x ),as I ⊆ Q . i i i i i i √ √ So, (Q : x ) = (I : x ) and by Lemma 2.8(iv), (I : x ) = (Q : x ) = P . Hence, {P , i i i i i i i 1 P , ... ., P }={Prime ideals P |∃x ∈ R such that P = (I : x)}. 2 n 3. Fuzzy Primary Decomposition of Fuzzy k-Ideals in Semirings In this section, we prove the existence and uniqueness of fuzzy primary decomposition of fuzzy k-ideals of semirings. We first recall some definitions from [22] which are required to prove the uniqueness Definition 3.1: Let μ be a fuzzy ideal of a semiring R. Then the radical of μ, denoted by μ is defined by μ(x) = sup {μ(x ), for all x ∈ R}. n≥1 Example 3.2: Consider the semiring R = (N, +,.) of non-negative integers with respect to usual addition and multiplication of integers. Define a fuzzy ideal μ of N as follows: n−1 n 1 − ,if x ∈ 2 \ 2 , n = 1, 2, ... μ(x) = . 1, if x = 0 1 2 n Observe that im μ ={0, 1, , , ... ., , ...} and clearly, 2 3 n+1 √ 1, if x ∈ μ(x) = . 0, if x ∈ Z \ Definition 3.3: Let μ be a non-constant fuzzy k-ideal of a semiring R. Then μ is said to be a fuzzy k-irreducible ideal of R if for any two fuzzy k-ideals θ and η of R, μ = θ ∩ η implies either μ = θ or μ = η. Example 3.4 ([22]): Let R be a semiring as in Example 3.2 and μ be a fuzzy ideal of R defined as follows: 1, if x ∈ μ(x) = . 0.4, otherwise Then, by Kar et al. [22, Theorem 4.2], one can easily check that μ is a fuzzy k- irreducible ideal of R. Definition 3.5: A fuzzy ideal μ of a semiring R is said to be fuzzy prime ideal of R if it is non-constant (i.e. |imμ|≥ 2) and any two fuzzy ideals θ and η of R, θ ◦ η ⊆ μ implies that either θ ⊆ μ or η ⊆ μ. Definition 3.6: A fuzzy ideal μ of a semiring R is said to be fuzzy primary ideal of R if it is non-constant (i.e. |imμ|≥ 2) and any two fuzzy ideals θ and η of R, θ ◦ η ⊆ μ implies that √ √ either θ ⊆ μ or η ⊆ μ.If μ = ν, then μ is called fuzzy ν-primary ideal of R. Definition 3.7: Let μ and θ be two fuzzy subsets of a semiring R. Then fuzzy colon ideal (μ : θ) is defined by (μ : θ)(x) = sup {λ(x) | λ ◦ θ ⊆ μ}, where IFS(R) denotes the set of λ∈IFS all fuzzy subsets of R. FUZZY INFORMATION AND ENGINEERING 229 Definition 3.8: Let μ be a fuzzy ideal of a semiring R. Then μ is defined as, μ ={x ∈ R| 0 0 μ(x) = μ(0)}. Definition 3.9: If μ is a fuzzy k-ideal of a semiring R, then μ is said to have a fuzzy primary decomposition if μ canbeexpressedas μ =∩ μ , where each μ is a fuzzy primary ideal i i i=1 of R. A fuzzy primary decomposition μ =∩ μ is said to be reduced if μ ’s are distinct i i i=1 and ∩ μ μ , for all i = 1, 2 ··· n. By Kar et al. [22, Theorem 5.3], a reduced fuzzy primary j i j=i decomposition can be obtained from a fuzzy primary decomposition μ =∩ μ , where i=1 each μ is a fuzzy primary ideal of R. The primary decomposition proved for k-ideals in a commutative Noetherian semiring (Theorem 2.5) can be generalised to fuzzy k-ideals as in [22, Theorem 5.2]. Thus we have Theorem 3.10 (Fuzzy Primary Decomposition of Fuzzy k-Ideals): Let R be an additively cancellative, yoked and commutative Noetherian semiring and μ a fuzzy k-ideal of R such that imμ ={1, α}, where α ∈ [0, 1). Then μ can be represented as a finite intersection of fuzzy primary ideals of R, i.e, μ has a primary decomposition. Similar to the primary decomposition of k-ideals in semirings, all μ ’s in the fuzzy primary decomposition of μ have an additional property that these are also fuzzy k-ideals. There are some errors (as mentioned below) in the result used for the proof of unique- ness of fuzzy primary decomposition of fuzzy k-ideals in semirings. So the question of establishing the result still remains unsolved. First, we state the result used for uniqueness. Theorem 3.11 ([22], Theorem 5.4): Let μ be a fuzzy k- ideal of a semiring R and μ =∩ μ i=1 be a reduced fuzzy primary decomposition of μ.Let λ be a fuzzy k-prime ideal of R. Then λ = μ forsomei = 1, 2 ··· .n if and only if there exists a fuzzy k-ideal θ of R such that θ μ and (μ : θ) = λ, i.e. (μ : θ) is a fuzzy λ-primary ideal of R. In the proof of above result, μ =∩ μ is a reduced fuzzy primary decomposition of i=1 √ √ fuzzy k-ideal μ,so μ ’s are distinct for each i = 1, 2, ... .n. But the authors have used μ = i i λ,for each i = 1, 2, ... .n in the step n n n n ∩ (μ : θ) = ∩ (μ : θ) = ∩ μ = ∩ λ = λ i i i i=1 i=1 i=1 i=1 contrary to the assumption that the decomposition is reduced. In order to establish a correct proof of the uniqueness of fuzzy primary decomposition of a fuzzy k-ideal in a semiring, first we need the following result: Lemma 3.12: Let μ be a fuzzy ν− primary ideal of a semiring R and θ afuzzy idealofR.Then the following hold: (i) If θ ⊆ μ, then (μ : θ) = χ , where χ is the chai function on R; R R (ii) If θ μ, then (μ : θ) is a fuzzy ν− primary ideal of R, i.e. (μ : θ) = ν. 230 R. P. SHARMA ET AL. Proof: (i) Let x ∈ R and θ ⊆ μ. Then we have (μ : θ)(x) = sup {λ(x)|λ ◦ θ ⊆ μ}. λ∈IFS(R) Also, (χ ◦ θ)(x) = sup{min(χ (y), θ(z))|x = yz} R R = sup{θ(z)|x = yz} ≤ θ(x) ≤ μ(x) as θ ⊆ μ. Thus, by the definition of colon ideal and (χ ◦ θ) ⊆ μ,weget (μ : θ) = χ . R R (ii) Let θ μ and η ◦ η ⊆ (μ : θ) 1 2 for some fuzzy ideals η and η of R. Then by the definition of fuzzy colon ideal, η ◦ η ◦ θ ⊆ 1 2 1 2 μ.Since μ is a fuzzy ν− primary ideal of R and θ μ,wehave η ◦ η ⊆ μ which further 1 2 √ √ √ implies that either η ⊆ μ or η ⊆ μ,as μ is a fuzzy prime ideal of R. 1 2 √ √ If η ⊆ μ, then by Kar et al. [22, Lemma 3.2 (iii) and Lemma 5.2 (i)], μ ⊆ (μ : θ) √ √ implies that η ⊆ (μ : θ). Similarly, if η ⊆ μ, then we get η ⊆ (μ : θ).Thus, (μ : θ) 1 2 2 is a fuzzy primary ideal of R. We now show that (μ : θ) = ν. For this, let x ∈ R. Then (μ : θ)(x) = sup{(μ : θ)(x )} n≥1 = sup sup {λ(x )|λ ◦ θ ⊆ μ} n≥1 λ∈IFS(R) = sup sup {λ(x )|λ ⊆ μ} n≥1 λ∈IFS(R) ≤ sup{ μ(x )} n≥1 √ √ = μ(x) = μ(x) √ √ as θ μ and μ is a fuzzy prime ideal of R.Thus, (μ : θ) ⊆ μ. Also by Kar et al. [22, √ √ √ √ Lemma 3.2 (iii) and Lemma 5.2 (i)], μ ⊆ (μ : θ) implies that (μ : θ) = μ = ν. Now, we give the correct proof of Theorem 3.11, which is required for the establishment of uniqueness of fuzzy primary decomposition of a fuzzy k- ideal of a semiring as follows: Theorem 3.13: Let R be an additively cancellative, yoked, zerosumfree and commutative Noetherian semiring, and μ a fuzzy k- ideal of R such that imμ ={1, α}, where α ∈ [0, 1).Let μ =∩ μ be a reduced fuzzy primary decomposition of μ and λ be a fuzzy prime ideal of R. i=1 Then λ = μ , for some i = 1, 2, ... ., n if and only if there exists a fuzzy k-ideal θ of R such that θ μ and (μ : θ) = λ, i.e. (μ : θ) is a fuzzy λ-primary ideal of R. FUZZY INFORMATION AND ENGINEERING 231 Proof: Let λ = μ , for some i = 1, 2, ... ., n.As μ =∩ μ is a reduced fuzzy k-primary i i i=1 decomposition of μ,so ∩ μ μ , for any i = j. Therefore, there exists some x ∈ R such j i i j=1 j=i n n that ∩ μ (x )>μ (x ). Assume that ∩ μ (x ) = a, where a ∈ [0, 1). We now establish a j i i i j i j=1 j=1 j=i j=i fuzzy subset θ of R as follows: a,if x ∈ θ(x) = . 0, otherwise By Lemma 3.8, the ideal x is a k-ideal as R is additively cancellative, yoked and zerosum- free, which clearly implies that θ is a fuzzy k-ideal of R. Also, n n θ(x ) = a = ∩ μ (x )>μ (x )> ∩ μ (x ) = μ(x ) i j i i i i i i j=1 i=1 j=i implies that θ μ and θ μ ,for i = j. Further, θ(x ) = a = ∩ μ (x ) = inf {μ (x ), μ (x ), ... ., μ (x ), μ (x ), ... .μ (x )}≤ μ (x ) i j i 1 i 2 i i−1 i i+1 i n i j i j=1 j=i for all j = i and j = 1, 2, ... n.Now,weclaimthat θ ⊆ μ , for all j = i and j = 1, 2, ... , n.Let x = x y ∈< x >, for some y ∈ R. Then i i i i θ(x) = θ(x ) ≤ μ (x ) ≤ μ (x y ) = μ (x) i j i j i i j implies that θ ⊆ μ for all j = i and j = 1, 2, ... , n. Thus, by Lemma 3.12 (i), we have (μ : θ) = χ , for all j = i, j = 1, 2, ... , n and (μ : θ) = μ = λ. i i Finally, by Kar et al. [22, Theorem 3.2 (v) and Lemma 5.2 (iii)], we get n n n (μ : θ) = ∩ μ : θ = ∩ (μ : θ) = ∩ (μ : θ) = (μ : θ) = λ. i i i i i=1 i=1 i=1 Also by Lemma 3.12 (ii), (μ : θ) is a fuzzy λ-primary ideal of R. Conversely, assume that there exists a fuzzy k-ideal θ of R such that θ μ and (μ : θ) = λ.As θ μ = ∩ μ,weget θ μ for some j ∈{1, 2 ··· .n} i j i=1 and by Lemma 3.12 (ii), we have (μ : θ) = μ . j j 232 R. P. SHARMA ET AL. Also, (μ : θ) = ∩ (μ : θ) = λ i=1 implies that ∩ (μ : θ) ∩ μ = λ. i j i=1 i=j Further, ⎛ ⎞ ⎛ ⎞ n n √ √ ⎝ ⎠ ⎝ ⎠ ∩ (μ : θ) ◦ μ ⊆ ∩ (μ : θ) ∩ μ = λ i j i j i=1 i=1 i=j i=j and n n λ = ∩ (μ : θ) ⊆ ∩ (μ : θ) i i i=1 i=1 i=j implies that μ ⊆ λ,as λ is prime. Also, n n λ = (μ : θ) = ∩ (μ : θ) = ∩ (μ : θ) ⊆ (μ : θ) = μ . i i j j i=1 i=1 Thus, we get λ = μ for some, j ∈{1, 2 ··· .n}. Finally, we are in a position to give the correct proof of uniqueness theorem of reduced fuzzy k-primary decomposition of a fuzzy k-ideal in semirings as follows: Theorem 3.14 (Uniqueness of Fuzzy Primary Decomposition): Let R be an additively can- cellative, yoked, zerosumfree and commutative Noetherian semiring, and μ a fuzzy k-ideal of R such that imμ ={1, α}, where α ∈ [0, 1).Let μ =∩ μ with μ = ν for i = 1, 2, ... .nand i i i i=1 μ =∩ ξ with ξ = η for i = 1, 2, ... .m be two reduced fuzzy k- primary decompositions i i i i=1 of μ. Then, n = mand {ν , ν ··· .ν }={η , η , ... ., η }. 1 2 n 1 2 m Proof: Let ν ∈{ν , ν ··· .ν }. Then by Theorem 3.13, there exists a fuzzy k-ideal θ of R such i 1 2 n that θ μ and (μ : θ) = u = ν for some, i ∈{1, 2, ... , n}. i i Also, since μ =∩ ξ with ξ = η for i = 1, 2, ... .m is another reduced fuzzy k-primary i i i i=1 decomposition of μ, therefore there exists some j ∈{1, 2 ··· .m} such that ν = (μ : θ) = ξ = η . This implies that j j {ν , ν ··· .ν }⊆{η , η , ... ., η } and so n ≤ m. 1 2 n 1 2 m By reversing the role of ν and ξ ,weget i j {η , η , ... ., η }⊆{ν , ν ··· .ν } and m ≤ n. 1 2 m 1 2 n Thus, {ν , ν ··· .ν }={η , η , ... ., η } and n = m. 1 2 n 1 2 m FUZZY INFORMATION AND ENGINEERING 233 4. Conclusions All the results that hold good for ideals in rings may not be true for the ideals in semirings and not even for the k-ideals in semirings. But this fact was ignored by various mathemati- cians while generalising the Lasker-Noether’s Theorem for semirings. This made Lescot [21] to establish weak primary decomposition for k-ideals in semirings in 2015. But the question of settling the primary decomposition for k- ideals still remained unsolved. In this paper, we prove: Let I be a k-ideal of an additively cancellative, yoked and commutative Noetherian semir- ing R. Then I can be uniquely represented as a finite intersection of primary ideals of R, i.e.I has a unique reduced primary decomposition. The above said result is also generalised for fuzzy k-ideals of same class of semirings. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on contributors Professor Ram Parkash Sharma has a remarkable 33 years of academic career (including 29 years in Himachal Pradesh University) with Doctorate from Ramanujan Institute for Advanced Study in Mathematics, University of Madras, India and has published 60 research papers in reputed jour- nals. Gold medallist in Masters, Prof. Sharma has been a dedicated and resolute researcher who has received highly prestigious fellowships, including UGC-JRF, Visiting Fellowship of Panjab University, Post-Doctoral Fellowship of NBHM, Department of Atomic Energy Govt. of India, SERC Visiting Fellow- ship, DST, Govt. India, Indian National Science Academy Fellowship, Hungarian Scholarship (Type-E) and most prestigious Commonwealth Fellowship (UK). Twelve Students have completed doctorate under his supervision. Dr. Madhu Dadhwal obtained her M.Sc. M.Phil. and Ph. D degrees from Himachal Pradesh University, Shimla, India. She was HPU Junior and Senior Research Fellow at the Department of Mathemat- ics, Himachal Pradesh University, Shimla, India. During 2210–2016, she worked at Department of Higher education Himachal (India) as Assistant Professor in Mathematics. From 2016 to till date, she is working at the Himachal Pradesh University, Shimla, India. Presently she is working as an Assistant Professor at the Department of Mathematics, Himachal Pradesh University, Shimla, India. Her main scientific interests are ternary semirings, group action and derivations in semirings, distinguishing labelling and distinguishing numbers for group actions. She is the author/co-author of more than 15 scientific research papers and 1 book. Richa Sharma obtained her M.Phil and Ph.D degrees from Himachal Pradesh University Shimla Himachal Pradesh, India. She was awarded HPU JRF and BSR fellowships. She was awarded gold medal in M.Phil. Presently she is working as an assistant professor at Government degree College Chintpurni Una India. She is the author/ co author of more than five research papers. Professor S. Kar obtained his M.Sc. and Ph. D degrees from the University of Calcutta, Kolkata, India. He was a CSIR Junior and Senior Research Fellow at the Department of Pure Mathematics, University of Calcutta, Kolkata, India. During 2005–2009, he worked at P. R. M. S. Mahavidyalaya, Bankura, West Bengal, India. From 2009 to till date, he is working at the Jadavpur University, Kolkata, India. Presently he is working as a Professor at the Department of Mathematics, Jadavpur University, Kolkata, India. His main scientific interests are ternary algebras, ordered algebras, ring derivation theory and algebraic graph theory. He is the author/coauthor of more than 60 scientific research papers and 3 books. He visited many places in abroad like Indonesia, Thailand, Vietnam, Hong Kong for academic activities. 234 R. P. SHARMA ET AL. ORCID Madhu Dadhwal http://orcid.org/0000-0002-6059-4408 References [1] Feng F, Zhao X, Jun YB. *-μ-semirings and *-λ-semirings. Theor Comput Sci. 2005;347:423–431. [2] Mahmood T, Tariq U. Generalized k-ideals in semirings using soft intersectional sets. Int J Algebra Statist. 2015;4(1):20–38. (ISSN:2314-4548) [3] Mahmood T, Shabir M. Characterizations of h-hemiregular and h-semisimple hemirings by interval valued fuzzy h-ideals. World Appl Sci J. 2012;17:1821–1827. (ISSN:1818-4952) [4] Shabir M, Anjum R. Characterizations of hemirings by the properties of their k-ideals. Appl Math (Irvine). 2013;4(5). Article ID: 31225, 16 pages. [5] Shabir M, Bashir S. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Apr 3, 2021
Keywords: Semirings; fuzzy k -ideals; primary ideals; primary decomposition; 16Y60
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