A Note on the Primary Decomposition of −Ideals in Semirings Ram Parkash Sharma 1, , Ricah Sharma 1, , S. Kar 2, and Madhu 1,∗ 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: mpatial.math@gmail.com a. rp_math_hpu@yahoo.com b. richasharma567@yahoo.com c. karsukhendu@yahoo.co.in ABSTRACT: We establish the primary decomposition and uniqueness of primary decomposition for −ideals in commutative Noetherian semirings. KEYWORDS: Semiring, −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 −ideals (if  +  ∈ ,  ∈ , then ∈ ) 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 −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 Noetherian ring, every ideal 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 −ideal can be represented as a finite intersection of −primary ideals [1, Theorem 4]. As observed in [4], there are some errors in the results used to prove this theorem. For example,  + ⁿ is not a −ideal, even if  is a −ideal. But the authors of [1] took it for granted that the ideal  + ⁿ is a −ideal. P. Lescot [4] found these errors after observing in Example 6.2 that {0} ideal may not be a finite intersection of −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 −ideal (other than {0} and semiring ) remained unsolved. In this direction, S. Kar et. al. [3, Theorem 4.4] proved that every proper − ideal of a commutative Noetherian semiring  can be expressed as a finite intersection of