A Note on the of 풌 −Ideals in Semirings

Ram Parkash Sharma1,푎, Ricah Sharma1,푏, S. Kar2,푐 and Madhu1,∗ ¹Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India ²Department of Mathematics, Jadavpur University, West Bengal, Kolkata-700032, India *Corresponding author E-mail address: [email protected] a. [email protected] b. [email protected] c. [email protected] ABSTRACT: We establish the primary decomposition and uniqueness of primary decomposition for 푘 −ideals in commutative Noetherian semirings. KEYWORDS: Semiring, 푘 −ideals, Primary ideals, Irreducible ideals, Primary decomposition. MSC: 16Y60, 16Y99. 1. Introduction Ideals play an important role in both semiring theory and ring theory, but in the absence of additive inverses in semirings, their structure differs from that of ring theory. Due to this difference in both the theories, the role of 푘 −ideals (if 푥 + 푦 ∈ 퐼, 푥 ∈ 퐼, then 푦 ∈ 퐼) becomes significant in semirings. It is pertinent to note here that many results which are true for ideals in rings have been established, by many authors, for 푘 −ideals in semirings (c.f. [5], [6], [7], [8]). This fact has motivated different researchers to settle the primary decomposition for k- ideals in semirings analogous to the primary decomposition theorem in rings (Lasker Noether theorem) which states that in a commutative , every can be described as a finite intersection of primary ideals. The above result of ring theory is not true for arbitrary ideals in semirings as noticed in [1]. R. E. Atani and S. E. Atani first proved that in a commutative Noetherian semiring, every proper 푘 −ideal can be represented as a finite intersection of 푘 −primary ideals [1, Theorem 4]. As observed in [4], there are some errors in the results used to prove this theorem. For example, 퐼 + 푅푎ⁿ is not a 푘 −ideal, even if 퐼 is a 푘 −ideal. But the authors of [1] took it for granted that the ideal 퐼 + 푅푎ⁿ is a 푘 −ideal. P. Lescot [4] found these errors after observing in Example 6.2 that {0} ideal may not be a finite intersection of 푘 −primary ideals in a commutative Noetherian semiring. With these observation, P. Lescot developed the theory of weak primary decomposition for semirings of characteristic 1. But still the question of settling the primary decomposition for a proper 푘 −ideal (other than {0} and semiring 푅) remained unsolved. In this direction, S. Kar et. al. [3, Theorem 4.4] proved that every proper 푘 − ideal of a commutative Noetherian semiring 푅 can be expressed as a finite intersection of 푘 −irreducible ideals, where an ideal 퐼 is 푘 −irreducible if for any two 푘 −ideals 퐽, 퐾 of 푅, 퐼 = 퐽 ∩ 퐾 implies that either 퐼 = 퐽 or 퐼 = 퐾. They tried to prove the primary decomposition by observing that in a commutative Noetherian semiring, every 푘 − is a 푘 − [3, Theorem 4.6]. But the proof of this result has the same errors again and the problem still remained unsolved. In this paper, we provide a correct proof of Theorem 4.6 of [3], which settle the primary decomposition for 푘 −ideals in commutative Noetherian semirings. In order to establish the uniqueness of primary decomposition for semirings, first we prove some basic results required for uniqueness and finally also establish the uniqueness of primary decomposition for 푘 −ideals of commutative Noetherian semirings, that is, if 퐼 = 푛 ⋂푖=1 푄푖, where each 푄푖 is a primary and √푄푖 = 푃푖, then the set {푃1, 푃2, … . 푃푛} is independent of particular choice of decomposition of 퐼.

2. Primary Decomposition of 풌 −Ideals in Semirings In this section, we prove the primary decomposition and uniqueness theorem for 푘 −ideals in commutative Noetherian semirings. First, we recall some definitions and results from [2] and [3] which are necessary to prove the main results of this section. Definition 2.1 [2]. A semiring is a nonempty set 푅 on which operations of addition and multiplication have been defined such that for 푟 ∈ 푅, the following conditions are satisfied:

(1) (푅, +) is a commutative monoid with identity element 0푅;

(2) (푅, . ) is a monoid with identity element 1푅; (3) Multiplication distributes over addition from either side;

(4) 0푅. 푟 = 0푅 = 푟. 0푅;

(5) 1푅 ≠ 0푅. A semiring 푅 is said to be commutative, if (푅, . ) is a commutative monoid. Throughout this paper, 푅 is a commutative semiring with identity.

Definition 2.2 [2]. Let 퐼 be an ideal of a semiring 푅. Then the radical of 퐼, denoted by √퐼 is defined as √퐼 = {푎 ∈ 푅| 푎ⁿ ∈ 퐼 for some 푛 ≥ 1}. Definition 2.3 [2]. A proper ideal 퐼 of a semiring 푅 is called a , if 푎푏 ∈ 퐼 for 푎, 푏 ∈ 푅 implies that either 푎 ∈ 퐼 or 푏 ∈ 퐼. Definition 2.4 [2]. A proper ideal 퐼 of a semiring 푅 is said to be a primary ideal of 푅 if for any 푥, 푦 ∈ 퐼, 푥푦 ∈ 퐼 implies that either 푥 ∈ 퐼 or 푦 ∈ √퐼. If √퐼 = 푃, then 퐼 is called 푃 −primary. Definition 2.5 [3]. A proper 푘 −ideal 퐼 of a semiring 푅 is called a 푘 −irreducible ideal of 푅 if for any two 푘 −ideals 퐽, 퐾 of 푅, 퐼 = 퐽 ∩ 퐾 implies that either 퐼 = 퐽 or 퐼 = 퐾. We now present the notion of primary decomposition of k-ideals as follows: Definition 2.6. Let 퐼 be a 푘 −ideal of a semiring 푅. Then 퐼 is said to have a primary 푛 decomposition if 퐼 can be expressed as 퐼 = ⋂푖=1 푄푖, where each 푄푖 is a primary ideal of 푅. 푛 Also, a primary decomposition of the type 퐼 = ⋂푖=1 푄푖 with √푄푖 = 푃푖, is called a reduced primary decomposition of 퐼, if 푃푖′푠 are distinct and 퐼 cannot be expressed as an intersection of a proper subset of ideals 푄푖 in the primary decomposition of 퐼. A reduced primary decomposition can be obtained from any primary decomposition by deleting those 푄푗 that 푛 contains ⋂푖=1 푄푖 and grouping together all distinct √푄푖′푠. 푖≠푗 The main aim of this paper is to prove the existence and uniqueness of primary decomposition for 푘 −ideals in a commutative Noetherian semiring. First, we prove the existence part as stated below: Theorem 2.7 (Primary Decomposition of 풌 −Ideals). Let 퐼 be a 푘 −ideal of a Noetherian semiring 푅. Then 퐼 can be represented as a finite intersection of primary ideals of 푅, that is, 퐼 has a reduced primary decomposition. The decomposition of 푘 −ideals in terms of 푘 −irreducible ideals has already been proved in [3] as follows: Theorem 2.8 [3, Theorem 4.4]. Every proper 푘 −ideal of a commutative Noetherian semiring 푅 can be expressed as a finite intersection of 푘 −irreducible ideals. Therefore, primary decomposition for 푘 −ideals in semirings follows immediately if we prove Theorem 2.9. Let 푅 be a commutative Noetherian semiring. Then every 푘 −irreducible ideal of 푅 is a primary ideal of 푅. Kar et. al. [3, Theorem 4.6] made an attempt to prove the same, but the proof of the result has some errors as mentioned in the introduction. Before we prove the required result, we analyse some common errors committed by many authors regarding 푘 −ideals, and prove some facts about k-ideals. First, we note that any ideal 퐼 =< 푎 > is not a 푘 −ideal in semirings as observed by J. S. Golan in [2, Example 6.17]. That is, if 퐼 = < (1 + 푥) > is an ideal of ℕ[푥] (semiring of polynomials over non-negative integers ℕ in the indeterminate 푥), then (1 + 푥)³ = (푥³ + 1) + 3푥(1 + 푥) ∈ 퐼, 3푥(1 + 푥) ∈ 퐼, but (푥³ + 1) ∉ 퐼 implies that 퐼 is not a 푘 −ideal of ℕ[푥]. Any ideal generated by a single element becomes a 푘 −ideal, if we impose some conditions on a semiring as shown below: Lemma 2.10. Let 푅 be an additively cancellative, yoked and zerosumfree semiring. Then for any 푎 ∈ 푅, the ideal 퐼 =< 푎 > is a 푘 −ideal of 푅. Proof. Let 푎₁ + 푎₂, 푎₁ ∈ 퐼. Then there exist some 푟₁, 푟₂ ∈ 푅 such that 푎₁ + 푎₂ = 푎푟₁ and 푎₁ = 푎푟₂. As 푅 yoked, so there exists some 푟₃ ∈ 푅 such that 푟₁ + 푟₃ = 푟₂ or 푟₂ + 푟₃ = 푟₁. If 푟₁ + 푟₃ = 푟₂, then 푎푟₁ = 푎푟₂ + 푎₂ = 푎푟₁ + 푎푟₃ + 푎₂ implies that 푎푟₃ + 푎₂ = 0, as 푅 is additively cancellative. Also, 푎₂ = 푎푟₃ = 0 ∈ 퐼, as 푅 is zerosumfree. Further, if 푟₂ + 푟₃ = 푟₁, then 푎푟₂ + 푎₂ = 푎푟₁ = 푎(푟₂ + 푟₃) = 푎푟₂ + 푎푟₃ implies that 푎₂ = 푎푟₃ ∈ 퐼, since 푅 is additively cancellative. Thus, 퐼 =< 푎 > is a 푘 −ideal of 푅.■ The semiring considered in above example is additively cancellative, zerosumfree, but it is not yoked, for let 푓(푥) = 5푥² + 9푥 + 2 and 푔(푥) = 11푥² + 3푥 + 5 be two polynomials in ℕ[푥], then there exists no ℎ(푥) ∈ ℕ[푥] such that either 푓(푥) + ℎ(푥) = 푔(푥) 표푟 푔(푥) + ℎ(푥) = 푓(푥). Similar to an ideal generated by a single element, the sum of two 푘 −ideals may not be a 푘 −ideal in a semiring. There are plenty of 푘 −ideals in the semiring ℕ, but their sum is not a 푘 −ideal. However, the sum of two 푘 −ideals is a 푘 −ideal in a lattice ordered semiring (c.f. [2], Corollary 21.22). While proving Theorem 2.9, the authors wrongly used that the ideals (푄+< 푎푘 >) and (푄+< 푏 >) are 푘 −ideals. In view of the above observations, Theorem 2.9 follows verbatim as proved in [3, Theorem 4.6] for additively cancellative, yoked and zerosumfree semirings, because in this case, both an ideal generated by a single element and sum of two 푘 −ideals is a 푘 −ideal. Here, we give a proof of Theorem 2.9 without resorting to these restrictions. Proof of Theorem 2.9. Let 푄 be a 푘 −irreducible ideal of a Noetherian semiring 푅. Let 푎푏 ∈ 푄 be such that 푏 ∉ 푄. Now, we construct two ideals 퐼 and 퐽 of 푅 as follows : 퐼 =< 푎푘 > +푄 and 퐽 =< 푏 > +푄. Then, clearly, 푄 ⊆ 퐼 ∩ 퐽. Let 푦 ∈ 퐼 ∩ 퐽. Then 푦 = 푎ⁿ푧 + 푞 for some 푧 ∈ 푅 and 푞 ∈ 푄. Again 푎퐽 ⊆ 푄 (since 푎푏 ∈ 푄) and so 푎푦 ∈ 푄 (since 푦 ∈ 퐽). Therefore, 푎푦 = 푎ⁿ⁺¹푧 + 푎푞. Thus, we get that 푎ⁿ⁺¹푧 + 푎푞 ∈ 푄. Also, 푎푞 ∈ 푄, since 푞 ∈ 푄 and 푄 is an ideal of 푅. Thus, it follows that 푎ⁿ⁺¹푧 ∈ 푄, since 푄 is a 푘 −ideal of 푅. Construct a set 퐴푛 = {푥 ∈ 푅 | 푎ⁿ푥 ∈ 푄}. It is easy to check that 퐴푛 is an ideal of 푅 and 퐴₁ ⊆ 퐴₂. . .. is an ascending chain of ideals. Since 푅 is Noetherian, 퐴푛=퐴푛+1=.... for some 푛 ∈ ℤ⁺. Again 푎ⁿ⁺¹푧 ∈ 푄 implies that 푧 ∈ 퐴푛+1 = 퐴푛. It demonstrates that 푎ⁿ푧 ∈ 푄 which implies that 푦 ∈ 푄. Thus, 퐼 ∩ 퐽 = 푄. Let 퐽푎푐(푅) denote the Jacobson radical of semiring 푅, that is, the intersection of all maximal 푘 −ideals of 푅. Assume that 퐴 is an ideal of 푅 and 퐽푎푐(퐴) denotes the intersection of all maximal 푘 −ideals of 푅 containing 퐴. If 퐴̅ denotes the 푘 −closure (i.e. 퐴̅ = {푎 ∈ 퐴} | 푎 + 푏 = 푐 for some 푏, 푐 ∈ 퐴) of an ideal 퐴 of 푅, we have (√퐼̅̅̅∩̅̅̅퐽) = √퐼̅ ∩ √퐽,̅ where √푃 denotes the nil 푃 (c.f. [4], Lemma 2.2). We know that √푃 ⊆ 퐽푎푐(푃). Thus, 퐼̅ ∩ 퐽̅ ⊆ √퐼 ̅ ∩ √퐽̅ = √퐼̅̅̅∩̅̅̅퐽 ⊆ 퐽푎푐(̅퐼̅̅∩̅̅̅퐽). Therefore, 퐽푎푐(퐼̅ ∩ 퐽)̅ ⊆ 퐽푎푐(̅퐼̅̅∩̅̅̅퐽). Again 퐼̅̅̅∩̅̅̅퐽 ⊆ 퐼̅ ∩ 퐽̅ ⇒ 퐽푎푐(̅퐼̅̅∩̅̅̅퐽) ⊆ 퐽푎푐(퐼̅ ∩ 퐽)̅ . Thus, 퐽푎푐(̅퐼̅̅∩̅̅̅퐽) = 퐽푎푐(퐼̅ ∩ 퐽)̅ = 퐽푎푐(퐼)̅ ∩ 퐽푎푐(퐽)̅ . So we have 퐽푎푐(̅퐼̅̅∩̅̅̅퐽) = 퐽푎푐(퐼)̅ ∩ 퐽푎푐(퐽)̅ . Now, 푄 = 퐼 ∩ 퐽 ⇒ 푄̅ = 퐼̅̅̅∩̅̅̅퐽 ⇒ 푄 = ̅ 퐼̅̅̅∩̅̅̅퐽, since 푄 is a 푘 −ideal of 푅. This shows that 퐽푎푐(푄) = 퐽푎푐(̅퐼̅̅∩̅̅̅퐽) = 퐽푎푐(퐼)̅ ∩ 퐽푎푐(퐽)̅ , that is, 퐽푎푐(푄) = 퐽푎푐(퐼)̅ ∩ 퐽푎푐(퐽)̅ . Again 푄 is a 푘 −irreducible ideal, which implies that 퐽푎푐(푄) is 푘 −irreducible. 퐽푎푐(푄) = 푄 ∩ 퐽푎푐(푅) but 퐽푎푐(푄) ≠ 퐽푎푐(푅). Thus, 퐽푎푐(푄) = 푄, since 퐽푎푐(푄) is 푘 −irreducible. Accordingly, 푄 = 퐽푎푐(퐼)̅ ∩ 퐽푎푐(퐽)̅ , where each of 퐽푎푐(퐼)̅ and 퐽푎푐(퐽)̅ are 푘 −ideals of 푅. Now 푏 ∈ 퐽 implies that 푏 ∈ 퐽푎푐(퐽)̅ , but 푏 ∉ 푄, that is, 푄 ≠ 퐽푎푐(퐽)̅ . So 푄 = 퐽푎푐(퐼)̅ , since 푄 is 푘 −irreducible. Now 푎ⁿ ∈ 퐼 ⇒ 푎ⁿ ∈ 퐽푎푐(퐼)̅ = 푄. Hence 푄 is 푘 −primary.■ Remark 2.11. Now Theorem 2.7 follows by combining [3, Theorem 4.4] and Theorem 2.9. It is important to note here that all 푄푖′푠 in the primary decomposition of a 푘 −ideal 퐼 have an additional property that these are also 푘 −ideals. Now it only remains to prove the uniqueness of primary decomposition of a 푘 −ideal 퐼. The following lemma will be used to prove the uniqueness of reduced primary decomposition of 푘 −ideals in semirings. Lemma 2.12. Let 푅 be a semiring and 푄 a 푃 −primary ideal of 푅. Then we have (i) If 푥 ∈ 푅, (푄: 푥) = {푟 ∈ 푅 | 푟푥 ∈ 푄} is an ideal of 푅; (ii) If 푥 ∈ 푄, then (푄: 푥) = 푅; (iii) If 푥 ∉ 푃, then (푄: 푥) = 푄; (iv) If 푥 ∉ 푄, then (푄: 푥) is 푃 −primary. Proof. (i) Let 푟₁, 푟₂ ∈ (푄: 푥) and 푎 ∈ 푅. Then 푟₁푥, 푟₂푥 ∈ 푄 implies that (푟₁ + 푟₂)푥 ∈ 푄 and 푟₁푎푥 ∈ 푄 as 푄 is an ideal of 푅. Thus, (푄: 푥) is an ideal of 푅. (ii) If 푥 ∈ 푄, then 푅푥 ⊆ 푄, as 푄 is an ideal of 푅 which implies that 푅 ⊆ (푄: 푥). Also, (푄: 푥) is an ideal of R and so (푄: 푥) = 푅. (iii) Assume that 푥 ∉ 푃. If 푎 ∈ 푄, then 푎푥 ∈ 푄 implies that 푎 ∈ (푄: 푥). For the converse part, suppose that 푏 ∉ 푄 and 푥푏 ∈ 푄. Then, 푥 ∈ √푄 = 푃 as 푄 is 푃 −primary ideal of 푅 which contradicts that 푥 ∉ 푃. Thus, 푥푏 ∉ 푄 and so 푏 ∉ (푄: 푥).

(iv) Suppose that 푥 ∉ 푄. If 푦 ∈ (푄: 푥), then 푥푦 ∈ 푄 implies that 푦 ∈ √푄 = 푃 as 푄 is 푃 −primary ideal of 푅. Thus, 푄 ⊆ (푄: 푥) ⊆ 푃 implies that 푃 = √푄 ⊆ √(푄: 푥) ⊆ √푃. Also, √푃 = 푃 , as 푃 is a prime ideal of 푅 and so 푃 = √(푄: 푥). Now, we show that (푄: 푥) is a primary ideal of 푅. The ideal (푄: 푥) is proper as 푥 ∉ 푄 and so, 1 ∉ (푄: 푥). Assume that 푎푏 ∈ (푄: 푥) and 푏 ∉ √(푄: 푥) for 푎, 푏 ∈ 푅. Then, 푎푏푥 ∈ 푄 and 푄 is a 푃 −primary ideal of 푅 which implies that either 푎푥 ∈ 푄 or 푏 ∈ 푃 = √(푄: 푥). Thus, 푎 ∈ (푄: 푥) as 푏 ∉ √(푄: 푥).■ We now prove the uniqueness of the reduced primary decomposition of a 푘 −ideal of a semiring as follows: Theorem 2.13 (Uniqueness of Primary Decomposition). Let 푅 be a commutative Noetherian 푛 semiring and 퐼 a 푘 −ideal of 푅. If 퐼 = ⋂푖=1 푄푖 is a reduced primary decomposition of 퐼 with

√푄푖 = 푃푖 for 𝑖 = 1,2. . . . 푛, then {푃₁, 푃₂. . . . 푃푛} = {Prime ideals 푃 | ∃ 푥 ∈ 푅 such that 푃 =

√(퐼: 푥)}. The set {푃₁, 푃₂. . . . 푃푛} is independent of the particular reduced primary decomposition chosen for 퐼.

푛 푛 푛 Proof. Let 푥 ∈ 푅. Then √(퐼: 푥) = √(⋂푖=1 푄푖 : 푥) = ⋂푖=1 √(푄푖: 푥) = ⋂ 푖=1 푃푖 by Lemma 2.12 푥∉푄푖 푛 (iv) and therefore, √(퐼: 푥) ⊆ 푃푖 for all 𝑖 = 1,2. . . . 푛. Also, if √(퐼: 푥) is prime, then ∏ 푖=1 푃푖 ⊆ 푥∉푄푖 푛 ⋂ 푖=1 푃푖 = √(퐼: 푥) implies that 푃푖 ⊆ √(퐼: 푥) for some 𝑖 = 1,2. . . 푛. Thus, we have {Prime 푥∉푄푖 ideals 푃 | ∃ 푥 ∈ 푅 such that 푃 = √(퐼: 푥)} ⊆ {푃₁, 푃₂. . . . 푃푛}. 푛 On the other hand, for 𝑖 ∈ {1,2. . . . 푛}, we have ⋂푗=1 푄푗 ⊈푄푖, as the primary decomposition 푗≠푖 푛 is reduced. So there exists some 푥푖 ∈ ⋂푗=1 푄푗 and 푥푖 ∈ 푄푖. If 푦 ∈ (푄푖: 푥푖), then 푦푥푖 ∈ 푄푖, and 푗≠푖 푛 푦푥푖 ∈ (⋂푗=1 푄푗) ∩ 푄푖 = 퐼 which implies that 푦 ∈ (퐼: 푥푖),. Thus, (푄푖: 푥푖) ⊆ (퐼: 푥푖) ⊆ (푄푖: 푥푖), 푗≠푖 as 퐼 ⊆ 푄푖. So (푄푖: 푥푖) = (퐼: 푥푖) implies that √(푄푖: 푥푖) = √(퐼: 푥푖) =푃푖 by Lemma 2.12 (iv).

Hence {푃₁, 푃₂. . . . 푃푛} = {Prime ideals 푃 | ∃ 푥 ∈ 푅 such that 푃 = √(퐼: 푥푖)}.■

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