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Ideals in Quotient Semirings Shahabaddin Ebrahimi Atani and Ameneh Gholamalipour Garfami Faculty of Mathematical Sciences, University of Guilan, P.O

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Ideals in Quotient Semirings Shahabaddin Ebrahimi Atani and Ameneh Gholamalipour Garfami Faculty of Mathematical Sciences, University of Guilan, P.O Chiang Mai J. Sci. 2013; 40(1) 77 Chiang Mai J. Sci. 2013; 40(1) : 77-82 http://it.science.cmu.ac.th/ejournal/ Contributed Paper Ideals in Quotient Semirings Shahabaddin Ebrahimi Atani and Ameneh Gholamalipour Garfami Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran. *Author for correspondence; e-mail: [email protected] Received: 10 November 2010 Accepted: 5 October 2011 ABSTRACT Since the class of quotient rings is contained in the class of quotient semirings, in this paper, we will make an intensive study of the properties of quotient semirings as compared to similar properties of quotient rings. The main aim of this paper is that of extending some well-known theorems in the theory of quotient rings to the theory quotient semirings. Keywords: quotient semirings, weakly prime ideals, weakly primal ideal, semidomain like semirings 1. INTRODUCTION Semirings are natural topic in algebra ideal as in the ring case. There are many to study because they are the algebraic different definitions of a quotient semiring structure of the set of natural numbers. appearing in the literature. P.J. Allen ([2]) Semirings also appear naturally in many introduced the notion of Q-ideal and a areas of mathematics. For example, semirings construction process was presented by are useful in the area of theoretical computer which one can build the quotient structure science as well as in the solution of of a semiring modulo a Q-ideal (also see problem in graph theory and optimiza- the results listed in [3-5,7,8]). If I is an tion. In structure, semirings lie between ideal of a semiring R, then Golan has semigroups and rings. The class of rings is presented the notion quotient semiring contained in the class of semirings [7,8]. R/I, but this definition is different from Therefore, all of the properties given here the definition of Allen (see Section 2). apply to rings. Here we follow the definition of Golan. This paper generalizes some well- The main part of this paper is devoted to known result on quotient rings in stating and proving analogues to several commutative rings to commutative well-known theorems in the theory of semirings. The maindifficulty is figuring quotient rings (see Section 2). out what additional hypotheses the In order to make this paper easier to semiring or ideal must satisfy to get similar follow, we recall here various notions results. Quotient semirings are determined from semiring theory which will be used by equivalence relations rather than by in the sequel. A commutative semiring R 78 Chiang Mai J. Sci. 2013; 40(1) is defined as an algebraic system (R,+,.) (r + I) + (s + I) = r + s + I and (r + I) + such that (R,+) and (R,.) are commutative (s + I) = rs + I [7]. Our starting point is semigroups, connected by a(b + c) = ab + ac the following lemma. for all a, b, c ∈ R, and there exists 0 ∈ R such that r + 0 = r and r0 = 0r = 0 for each Lemma 2.1 Let I be an ideal of a semiring R. r ∈ R. In this paper all semirings considered Then the following hold: will be assumed to be commutative with (1) If a ∈ I, then a + I = I. non-zero identity. A subset I of a semiring (2) If I is a k-ideal of R and a ∈ I, then R will be called an ideal if a, b ∈ I and a + I = b + I for every b ∈ R if and only if r ∈ R implies a + b ∈ I and ra ∈ I. b ∈ I. In particular, c + I = I if and only if A subtractive ideal (= k-ideal) J is an ideal c ∈ I. such that if x, x + y ∈ J then y ∈ J (so {0} is a k-ideal of R). A prim ideal of R is a Proof. (1) Since a + 0 = 0 + a, we conclude proper ideal P of R in which x ∈ P or that a ~ 0; hence a + I = 0 + 1. y ∈ P whenever xy ∈ P. A primary ideal (2) Let a + I = b + I for every b ∈ R. P of R is a proper ideal of R such that, if Then a + u = b + v for some u, v ∈ I; so xy ∈ P and x ∉ P, then y ∈ P = {r ∈ R : rn ∈ P b ∈ I since I is a k-ideal. The other for some positive integer n}. A semiring R implication follows from (1) and the fact I is said to be a semidomain if ab = 0 (a, b ∈ R), is a k-ideal of R. then either a = 0 or b = 0. A semifield is a semiring in which non-zero elements from Lemma 2.2 Let I and J be ideals of a a group under multiplication. An element semiring R with I ⊆ J. Then the following a of a semiring R is called zero-divisor of hold: R if there exists 0 ≠ b ∈ R such that ab = 0 (1) J/I ={a + I: a ∈ J} is an ideal of (not here that we include 0 in the set of R/I. In particuar, if J is a k-ideal of R, then zero-divisors of semiring). The collection J/I is a k-ideal of R/I. of all zero-divisors of R will be denoted by (2) If 1+I ∈ J/I, then R/I= J/I. Z(R). Furthermore, the subset {a ∈ R: (3) If a+I is a invertible element of R/I there exists a positive integer such that with a+I ∈ J/I, then R/I=J/I. an = 0} of Z(R) consisting of the nilpotent elements of R will be denoted by nil (R), Proof. (1)Cleary, 0 + I ∈ J/I. Let a + I, b + the nilradical of R. I ∈ J/I and r + I ∈ R)/I. It is easy to see that (a + I) + (b + I) = a + b + I ∈ J/I and (r + I) 2. RESULTS (a + I) = ra + I ∈ J/I. Thus J/I is an ideal Quotients semirings are determined of R/I. Finally, assume that u + I ∈ J/I and by equivalence relations rather than by (u + I) + (v + I) = u + v + I ∈ J/I, where ideals as in the ring case. If I is an ideal of u ∈ J and v ∈ R. It then follows that semiring R, we define a relation on R, given u + v + t1 = c + t2 for some t1, t2 ∈ I and c ∈ J; by r1 ~ r2 if and only if there exist a1, a2 ∈ I hence v ∈ J since J is a k-ideal. Thus v + I satisfying r1 + a1 = r2 + a2. Then is an ∈ J/I, and the proof is complete. equivalence relation on R, and we denote (2) Let x + I ∈ R/I. Then (x + I) (1 + I) the equivalence class of r by r + I and these = x + I ∈ J/I; hence R/I ⊆ J/I, as required. collection of all equivalence classes by R/I. (3) Follows from (2). Golan shows that R/I is a semiring with Chiang Mai J. Sci. 2013; 40(1) 79 Theorem 2.3 Let I be an ideal of a semiring Theorem 2.4 Let P be a proper k-ideal of R. Then the following hold: a semiring R. Then the following hold: (1) If L is an ideal of R/I, then L = J/I for (1) P is a maximal k-ideal of R if and some ideal J of R. only if R/I is a semifield. (2) If P is a k-ideal of R with I ⊆ P, then (2) If I is an ideal of R with I ⊆ P, then P is a prime ideal of R if and only if P/I is a P is a maximal k-ideal of R if and only if prime ideal of R/I. P/I is a maximal ideal of R/I. (3) I is a prime k-ideal of R if and only R/I (0) if is a semidomain. In particular, is Proof. (1) Let P be a maximal ideal of R. prime if and onlyif only R is a semidomain. It suffices to show that every non-zero Proof. (1) Assume that J = {r ∈ R : r +I ∈ L} element a + P of R/P is invertible. By and let a ∈ I. Then by Lemma 2.1, a + I = Lemma 2.1, a ∉ P; hence P + Ra = R by 0 + I ∈ L; hence I ⊆ J. Let a,b ∈ J and r ∈ R. maximality of P. There exist r ∈ R and p ∈ P Then (a + I)+(b + I) = a + b + I ∈ L; so such that ra + p = 1. It then follows from a + b ∈ J. Similarly, ra ∈ J. Thus J is an Lemma 2.1 that (r + P)(a + P) = 1 + P. ideal of R. Finally, it is easy to see that Thus a + P is invertible. Conversely, L = J/I. assume that R/P is a semifield and P J (2) Let P be a prime ideal of R. Suppose for some k-ideal J of R; we show that J = R. that r + I, s + I ∈ P/I are such that (r + I) Then there is an element b ∈ J-P such that (s + I) = rs + I ∈ P/I, where r, s ∈ R. Then b + P is invertible in R/P, so (b+P)(c+P) = rs + I = P + I for some p ∈ P. This implies bc + P = 1+P forsome c + P ∈ R/P. that rs ∈ P since P is a k-ideal.
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