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Some Remarks on Semirings and Their Ideals 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).
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