SOME REMARKS ON SEMIRINGS AND THEIR IDEALS PEYMAN NASEHPOUR Abstract. In this paper, we give semiring version of some classical results in commutative algebra related to Euclidean rings, PIDs, UFDs, G-domains, and GCD and integrally closed domains. 0. Introduction The ideal theoretic method for studying commutative rings has a long and fruitful history [20]. The subject of multiplicative ideal theory is, roughly speaking, the description of multiplicative structure of a ring by means of ideals or certain systems of ideals of that ring [17]. Multiplicative ideal theory, originated in the works of Dedekind [5], Pr¨ufer [33] and Krull [24], is a powerful tool both in commutative algebra [9,27] and in a more general context, in commutative monoid theory [1,17]. Semirings are ring-like algebraic structures that subtraction is either impossible or disallowed, interesting generalizations of rings and distributive lattices, and have important applications in many different branches of science and engineering [14]. For general books on semiring theory, one may refer to the resources [10–13,16,19]. Since different authors define semirings differently, it is very important to clarify, from the beginning, what we mean by a semiring in this paper. By a semiring, we understand an algebraic structure, consisting of a nonempty set S with two operations of addition and multiplication such that the following conditions are satisfied: (1) (S, +) is a commutative monoid with identity element 0; (2) (S, ·) is a commutative monoid with identity element 1 6= 0; (3) Multiplication distributes over addition, i.e., a · (b + c)= a · b + a · c for all a,b,c ∈ S; (4) The element 0 is the absorbing element of the multiplication, i.e., s · 0=0 for all s ∈ S. arXiv:1804.00593v2 [math.AC] 19 Nov 2018 Some of the topics in multiplicative ideal theory for commutative rings have been generalized and investigated for semirings [8, 21, 26, 28–32]. Also see Chapter 7 of the book [12]. The purpose of this paper is to generalize some other concepts of multiplicative ideal theory in commutative rings and investigate them in semirings. Let us recall that a nonempty subset I of a semiring S is said to be an ideal of S, if a + b ∈ I for all a,b ∈ I and sa ∈ I for all s ∈ S and a ∈ I [2]. An ideal I of a semiring S is called a proper ideal of the semiring S, if I 6= S. A proper ideal P of a semiring S is called a prime ideal of S, if ab ∈ P implies either a ∈ P or b ∈ P . We collect all the prime ideals of a semiring S in the set Spec(S). Note that 2010 Mathematics Subject Classification. 16Y60, 13A15. Key words and phrases. Euclidean semirings, Unique factorization semidomains, Principal ideal semidomains, Integrally closed semidomains, Goldman-Krull semidomains. 1 2 PEYMAN NASEHPOUR a nonzero and nonunit element s of a semiring S is said to be irreducible if s = s1s2 for some s1,s2 ∈ S, then either s1 or s2 is unit (multiplicatively invertible). An element p ∈ S is said to be a prime element, if the principal ideal (p) is a prime ideal of S. Also note that a semiring S is semidomain if ab = ac implies b = c for all b,c ∈ S and all nonzero a ∈ S. A semidomain S is called to be a unique factorization semidomain (for short UFSD) if the following conditions hold: UF1 Each irreducible element of S is a prime element of S. UF2 Any nonzero, nonunit element of S is a product of irreducible elements of S. Section 3 of the paper is devoted to unique factorization semidomains. Beside some basic but interesting results, in Theorem 3.7, we show that a semidomain S is a UFSD if and only if every nonzero prime ideal of S contains a prime element. The ring version of this result is due to I. Kaplansky [22]. Similar to the concept of field of fractions in ring theory, one can define the semifield of fractions F (S) of the semidomain S [11, p. 22]. Note that if S is a semidomain and F (S) its semifield of fractions, then by definition [6, p. 88], an element u ∈ F (S) is said to be integral over S if there exist n n−1 n−1 a1,...,an and b1,...,bn in S such that u + a1u + ··· + an = b1u + ··· + bn. The semidomain S is said to be integrally closed if any integral element of F (S) over S belongs to S. We also mention that a nonempty subset W of a semiring S is said to be a multiplicatively closed set (for short an MC-set) if 1 ∈ W and for all w1, w2 ∈ W , we have w1w2 ∈ W . Note that if U is an MC-set of a semiring S, one can define the localization of S at U, similar to the definition of the localization in ring theory (refer to [23] and [12, §11]). Section 4 is devoted to Integrally closed semirings and similar to ring theory, in Theorem 4.2, for a semidomain S, we prove that the following statements are equivalent: (1) S is an integrally closed semidomain. (2) ST is an integrally closed semidomain for any MC-set T of S. (3) Sm is an integrally closed semidomain for any maximal ideal m of S. In section 5, we generalize the concept of G-domains [22, p. 12] and define a semidomain S to be a Goldman-Krull semidomain if its semifield of fractions F (S) is a finitely generated semiring over S. In Corollary 5.6, we show that a semidomain S is a Goldman-Krull semidomain if and only if \ P 6= (0). P ∈Spec(S)−{0} Let us recall that an ideal I of a semiring S is subtractive if a + b ∈ I and a ∈ I imply that b ∈ I for all a,b ∈ S. An ideal I of a semiring S is called principal if I = {sa : s ∈ S} for some a ∈ S. Such an ideal I of S is denoted by (a). A semiring S is called a principal ideal semidomain (for short PISD) if S is a semidomain and each ideal of S is principal. In Theorem 5.7, we prove that a PISD is a Goldman-Krull semidomain if and only if it has only finitely many prime ideals. Sections 1 and 2 are devoted to Euclidean and principal ideal semirings. These semirings provide some interesting examples for the other facts that we bring in the rest of the paper. SOMEREMARKSONSEMIRINGSANDTHEIRIDEALS 3 1. Euclidean Semirings Let us recall that by definition, a totally ordered set (O, ≤) is said to be well- ordered, if every nonempty subset of O has a least element [3, p. 38]. In the following, we clarify what we mean by a Euclidean semiring in this paper: Definition 1.1. Let (O, ) be a well-ordered set with no greatest element. We annex an element +∞ to O and denote the new set by O∞ and define x +∞ for any element x ∈ O∞. An O∞-Euclidean norm on a semiring S is a function δ : S → O∞ with the following properties: (1) δ(s)=+∞ if and only if s = 0 for all s ∈ S. (2) If a,b ∈ S with b 6= 0, then there are elements q, r ∈ S such that a = bq + r with either r =0, or δ(r) ≺ δ(b). In such a case, we say that S is an O∞-Euclidean semiring. Remark 1.2. As far as the author knows, the first resource defining some kinds of Euclidean semirings is the paper [4]. For more on “Euclidean semirings”, one can refer to [18] and [12, §12]. We also note that most algebra texts on rings and semirings require an N∞-Euclidean norm to have the following additional property: • δ(b) δ(sb), for all b,s ∈ S. In Proposition 1.3, which is a generalization of Proposition 12.10 in [12], we show that this condition is superfluous. Proposition 1.3. Let δ be an O∞-Euclidean norm on a semiring S. Then, there ∗ exists an O∞-Euclidean norm δ on S with the following properties: (1) δ∗(a) δ(a), for all a ∈ S. (2) δ∗(b) δ(sb), for all b,s ∈ S. ∗ ∗ Proof. Define δ (a) = min{δ(sa): s ∈ S}. Clearly, δ : S → O∞ satisfies the ∗ conditions (1) and (2). Now, we prove that δ is an O∞-Euclidean norm on S. It is straightforward to see that δ(s)=+∞ if and only if s = 0 for all s ∈ S. Let a,b ∈ S with b 6= 0. Clearly, there is an s ∈ S, such that δ∗(b)= δ(sb). Since δ is an O∞-Euclidean norm on the semiring S, there exist elements q and r in S such that a = qsb + r, with either r = 0, or δ(r) ≺ δ(sb). If r 6= 0, we have δ∗(r) δ(r) ≺ δ(sb)= δ∗(b) and the proof is complete. Proposition 1.4. Let S be an O∞-Euclidean semiring. Then, each subtractive ideal of S is principal. Proof. Let I be a nonzero subtractive ideal of S and set M = {δ(s): s ∈ I}. Obviously M ⊆ O and by definition M has a -least element. Let δ(b) be the -least element of M, where b ∈ I. It is clear that the principal ideal (b) is a subset of I. Our claim is that I ⊆ (b). Now, take a ∈ I −{0}. So, there exist elements q, r ∈ S such that a = bq + r with either r = 0 or δ(r) ≺ δ(b).