Ideal Theory in Graded Semirings -.:: Natural Sciences Publishing

Appl. Math. Inf. Sci. 7, No. 1, 87-91 (2013) 87

Applied Mathematics & Information Sciences An International Journal

°c 2013 NSP Natural Sciences Publishing Cor.

Ideal theory in graded semirings

P. J. Allen1, H. S. Kim2 and J. Neggers1 1 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U.S.A. 2 Department of Mathematics, Hanyang University, Seoul 133-791, Korea

Received: 12 Jun 2012; Revised 2 Sep. 2012 ; Accepted 15 Sep. 2012 Published online: 1 Jan. 2013

Abstract:√ An A-semiring has commutative multiplication and the property that every proper ideal B is contained in a prime ideal P , with B, the intersection of all such prime ideals. In this paper, we deﬁne homogeneous√ ideals and their radicals√ in a graded semiring R. When B is a proper homogeneous ideal in an A-semiring R, we show that B is homogeneous whenever B is a k-ideal. We also give necessary and sufﬁcient conditions that a homogeneous k-ideal P be completely prime (i.e., F 6∈ P,G 6∈ P implies FG 6∈ P ) in any graded semiring. Indeed, we may restrict F and G to be homogeneous elements of R.

Keywords: semiring, (k-) ideal, homogeneous (ideal), graded.

1. Introduction A set R together with two associative binary operations called addition and multiplication (denoted by + and The notion of semiring was first introduced by H. S. Van- ·, respectively) will be called a semiring provided: diver in 1934, and since then many other researchers also developed the theory of semirings as a generalization of (i)addition is a commutative operation; rings. Semirings occur in different mathematical fields, e.g., (ii)there exist 0 ∈ R such that x+0 = x and x0 = 0x = 0 as ideals of a ring, as positive cones of partially ordered for each x ∈ R, and rings and fields, in the context of topological considera- (iii)multiplication distributes over addition both from the tions, and in the foundations of arithmetic, including ques- left and from the right. tions raised by school education ([6]). In the 1980’s the The element 0 in item (ii) is called the zero of the theory of semirings contributed to computer science, since semiring R. A subset S of the semiring R will be called the rapid development of computer science needed addi- a subsemiring of R provided (1) x + y ∈ S and xy ∈ S tional theoretical mathematical background. The semiring whenever x, y ∈ S and (2) 0 ∈ S. By an additive sub- structure does not contain an additive inverse, and this point semigroup, we mean a subset of R that contains the zero is very helpful in developing the theoretical structure of of R and is closed under addition. A subset I of a semir- computer science. For example, hemirings, as a semiring ing R will be called an ideal if a, b ∈ I and r ∈ R implies with zero and commutative addition, appeared in a natu- a + b ∈ I, ra ∈ I and ar ∈ I. ral manner in some applications to the theory of automata and formal languages. Recently, J. S. Han and et al. ([5]) discussed semiring orders in semirings. We refer to J. S. Golan’s remarkable book for general reference ([4]). 3. Graded semirings Whenever R is a commutative semiring, we let R[X] de- 2. Preliminaries note the semiring of polynomials with coefficients in R over the transcendental element X. Then each non-zero element f(X) in R[X] can be represented by a unique sum There are many different definitions of a semiring appear- P k ni ni ing in the literature. Throughout this paper, a semiring will i=1 ani X of non-zero monomials ani X of unique be defined as follows: degrees. An ideal B in R[X] is said to be homogeneous

∗ Corresponding author: e-mail: [email protected]; [email protected]; [email protected]

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P k ni ni if f(X) = i=1 ani X ∈ B implies ani X ∈ B, for element x in R can be written, in a unique way, as a fi- each i. In this article, a number of properties of homoge- nite sum of non-zero homogeneous elements of distinct neous ideals will be derived. However, it will not be nec- degrees. P essary to restrict ourselves to polynomial semirings since Definition 3.5. Let R = q∈Z Rq be a graded semir- homogeneous ideals can be studied in semirings more gen- Pk ing and let x be a non-zero element in R. If i=1 rqi is the eral than polynomial semirings, namely in graded semir- decomposition of x into non-zero homogeneous elements ings. The first problem encounted will be the presentation of distinct degree, the homogeneous elements rq1 , rq2 , ··· , rqk of a definition of a graded semiring. will be called the homogeneous components of x, and the Definition 3.1. Let {Rq}q∈Z be a collection of addi- homogeneous component of x of least degree will be called tive subsemigroups of a semiring R. We say that R is the the initial component of x. internal direct sum of the collection {Rq}q∈Z provided: Let R be a semiring and let R[X] be the semiring of (1)Each non-zero element x in R has a unique represen- polynomials over R. If q is a non-negative integer, then Pk R tation of the form rq where qi 6= qj if i 6= j and q will denote the set consisting of the zero polynomial i=1 i O(X) together with all basic polynomials of degree q. If rqi is a non-zero element in Rqi for each i; and q is a negative integer, Rq will denote the set {O(X)}. It (2)If rq1 , rq2 , ··· , rqk are non-zero elements in R where Pk is clear that {R } is a collection of additive subsemi- q 6= q if i 6= j, and each r ∈ R , then r is q q∈Z i j qi qi i=1 qi groups of the semiring R[X]. With the aid of this notation, a non-zero element in R. P it can be shown that every polynomial semiring is a graded The notation “R = R ” will be used to indi- q∈Z q semiring. P cate that the semiring R is the internal direct sum of the Theorem 3.6. If R is a semiring , then R[X] = q∈Z Rq collection {R } . Although the collection {R } in q q∈Z q q∈Z is a graded semiring, where the collection {Rq}q∈Z is de- the above definition could be taken with Z any non-empty fined as above. set, we will always use Z to denote the ring of integers. Proof. It is easy to observe that R[X] is the internal Let Z be denote the ring of integers and let {Rq}q∈Z q q0 direct sum of the collection {Rq}q∈Z . If aqX and bq0 X be a collectionP of additive subsemigroups of a semiring are basic polynomials of degree q and q0, respectively, then R. The symbol q∈Z Rq will denote the set consisting of q q0 aqX · bq0 X is either O(X) or a basic polynomial of 0 together with all x in R for which there exist integers 0 0 degree q+q . Thus, R R 0 ⊂ R 0 for each q, q ∈ Z. An q , q , ··· , q , (q 6= q if i 6= j), and non-zero elements q q q+q 1 2 k i j P inspection of the definition of addition in R[X] shows that r , r , ··· , r r ∈ R x = k r q1 q2 qk such that qi qi and i=1 qi . condition (3) in Definition 3.2 is satisfied, and the proof is Definition 3.2. A semiring R is said to be graded if complete. ut there exists a collection {Rq}q∈Z of additive subsemigroups The following example will show that there exist graded of R satisfying the following conditions: P semirings other than polynomial semirings, and it will be clear that the notion of a graded semiring is a generaliza- (1)R = q∈Z Rq; 0 tion of the notion of a polynomial semiring. (2)RqRq0 ⊂ Rq+q0 for each q, q ∈ Z; Example 3.7. Let R be a semiring. Let R0 = R and let (3)If rq1 , rq2 , ··· , rqk are non-zero elements in R where Pk R = {0} if q is a non-zero element in Z. It is clear that q 6= q if i 6= j, and each r ∈ R , then r is q i j qi qi i=1 qi {R } is a collection of additive subsemigroups of the a non-zero element in R. q q∈Z P semiring R. An inspection shows that R = q∈Z Rq is a P Whenever R is a graded semiring, the notation “R = graded semiring. If R has only a finiteP number of elements, q∈Z Rq” will be used to indicate that {Rq}q∈Z is the then the finite, graded semiring R = q∈Z Rq cannot be a given collection of additive subsemigroups satisfying the polynomial semiring, since polynomial semirings contain above definition. P an infinite number of elements. Theorem 3.3. If R = q∈Z Rq is a graded semiring, and if n and m are distinct integers, then Rn ∩ Rm = ∅ or Rn ∩ Rm = {0}, where 0 denotes the zero in R. 4. Homogeneous ideals in graded semirings Proof. Assume there exists a non-zero element x in R x ∈ R ∩ R x ∈ R x ∈ R such that n m. Thus, n and m and it DefinitionP 4.1. Let B be an ideal in the graded semiring follows that x does not have a unique representation of the R = q∈Z Rq. If the homogeneous components of each Pk form i=1 rqi . Consequently, R is not the internal direct non-zero element in B belong to B, then B will be called sum of the collection {Rq}q∈Z , a contradiction. Note that a homogeneous ideal. it may happen that Rn ∩ RmP= ∅. ut Theorem 4.2. If B is an ideal in the commutative semir- Definition 3.4. Let R = q∈Z Rq be a graded semiring R, then B[X]Pis a homogeneous ideal in the graded ing. An element of R is said to be homogeneous if it be- semiring R[X] = q∈Z Rq. ∞ longs to an Rq, and it is said to be homogeneous of degree X n q if it belongs to Rq and is different from zero. Proof. Let f(X) = anX be a non-zero polyno- Since the graded semiring R is the internal direct sum Pn=0 k qi of the collection {Rq}q∈Z , it is clear that every non-zero mial in B[X] and let i=1 aqi X be the decomposition

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of f(X) into its homogeneous components. Clearly, f(X) ∈ for each i and j. Moreover,

B[X] implies an ∈ B ∪ {0} for each non-negative integer n n mj qi X X X n. Consequently, aq X ∈ B[X] i = 1, 2, ··· , k, and it j i p = rjbj = rj( bi ) follows that B[X] is a homogeneous ideal in R[X]. ut j=1 j=1 i=1 Further examples of homogeneous ideals can be con- Xn Xmj structed as follows: j = ( rjbi ). B R Example 4.3. Let beP an ideal in the semiring . In j=1 i=1 view of Example 3.7, R = q∈Z Rq is a graded semiring, Consequently, p is a linear combination in Φ∗. Thus, if B and it is clear that B is a homogeneousP ideal in RP. 0 0 has a basis, then the fact that B is homogeneous implies Definition 4.4. Let R = Rq and R = Rq q∈Z q∈Z that B has a basis consisting of homogeneous elements. be two graded semirings. A homomorphism η of R into R0 Conversely, suppose that B has a basis Φ consisting is said to be homogeneous of degree s if η(R ) ⊂ R0 q q+s of homogeneous elements. Let p be a non-zero element for each q ∈ Z. Pk in B and let pq be the decomposition of p into its Theorem 4.5. Let η be a homogeneousP homomorphism i=1 i homogeneous components. It will be shown that pqi ∈ B of the graded semiring R = q∈Z Rq into the graded 0 P 0 for each i, and it will follow that B is homogeneous. Since semiring R = q∈Z Rq . If ker(η) 6= ∅, then ker(η) is a Φ is a basis for B, there exist elements r , r , ··· , r in homogeneous ideal in R. 1 2 n R andP there exist elementsP b1, b2, ··· , bn in Φ such that Proof. Suppose η is a homogeneous homomorphism n ni i p = i=1 ribi. Let j=1 rj be the decomposition of ri of degree s. Let p be a non-zero element in ker(η) and into its homogeneous components. Clearly, Pk let rq be the decomposition of p into homogeneous i=1 i Xn Xn Xni elements of distinct degrees. Since p ∈ ker(η), it is clear i p = ribi = ( rj )bi that i=1 i=1 j=1 Xk Xk Xn Xni rq η = ( rq )η = pη = 0. i i i = ( rj bi), i=1 i=1 i=1 j=1 Since η is a homogeneous homomorphism of degree s, it i and rj bi is a homogeneous element in R. Thus, the above follows that rqi η ∈ Rqi+s, i = 1, 2, ··· , k. Condition (3) sum is the sum of homogeneous elements. If all terms of of Definition 2 implies r η = 0, i = 1, 2, ··· , k. Thus, qi the sameP degreeP are combined into one term, it is clear the homogeneous components of p belong to ker(η) and n ni i that ( rj bi) is the decomposition of p into its the proof is complete. ut i=1 j=1 homogeneous components. Consequently, each pqi must With the aid of the following definitions, a necessary be a linear combination in Φ. Thus, each pqi ∈ B. ut and sufficient condition can be given for an ideal to be ho- Theorem 4.8 indicates a method for generating homo- mogeneous in a graded semiring. geneous ideals in a graded semiring with an identity. In Definition 4.6. Let Φ be a set of non-zero elements view of Theorem 4.2, Example 4.3 and the above result, it in the semiring R.A linear combination in Φ is a sum is clear that the homogeneous ideals form a large class of Pk i=1 ribi where r1, r2, ··· , rk are elements in R and b1, b2, ···ideals, bk in graded semirings. are elements in Φ. The following theorem will show that the class of ho- Definition 4.7. Let B be an ideal in the semiring R.A mogeneous ideals in a graded semiring with zero is closed subset Φ of B will be called a basis for B if B is the set of under the standard ideal-theoretic operations. all linear combinations in Φ. Theorem 4.9. Let A and B be ideals in a graded semir- Theorem 4.8. Let R be a graded semiring and let B ing R. If A and B are homogeneous, then A+B and A∩B be an ideal in R. If B has a basis, then B is homogeneous are homogeneous ideals in R. if and only if B has a basis consisting of homogeneous Proof. It is clear that A + B and A ∩ B are ideals in R. Let p be a non-zero element in A + B. Thus, p = a + b elements. P P a ∈ A b ∈ B p a p b Proof. Suppose B is a homogeneous ideal and let Φ be where and . Let qi and qj be the decompositions of a and b, respectively, into their homo- a basis for B. It is clear that Φ ⊂ B. If x ∈ Φ, let Φx denote the set of homogeneous component of x, and let Φ∗ = geneous components. Since A and B are homogeneous, it is clear that each p a ∈ A and each p b ∈ B. Clearly, ∪x∈ΦΦx. Since B is homogeneous, it follows that Φx ⊂ B qi qj ∗ for each x ∈ Φ. Thus, Φ is a set of homogeneous elements p = a + b ∗ X X X and Φ ⊂ B. Since B is an ideal, any linear combination a b ∗ = p + p = p , in Φ is an element in B. Let p be an element in B. Since qi qj qm Φ is a basis for B, there exist elements r1, r2, ··· , rn in where the terms of like degreeP are combined to form a sin- R and there exist elements b1, b2, ··· , bn in Φ such that p Pn Pmj j gle term in the last sum. Thus, qm is the decomposition p = j=1 rjbj. Let i=1 bi be the decomposition of bj of p into its homogeneous components, and it is clear that j ∗ into homogeneous components. It is clear that bi ∈ Φ , each pqm ∈ A + B. Thus, A + B is a homogeneous ideal

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in R. The assertion relative to A ∩ B results trivially from sum X X X X the deﬁnition of homogeneous ideal. ut ( fi)( gj) + ( fi)( gj)+

In [1], the authors defined and investigated several no- ij0 proper ideal in an R, the radical of B is denoted by B and is defined to be the intersection of all prime ideals in which is in turn equal to FG an element in P . Since the R that contain B. first three terms of the above sum are elements in P , and An ideal B in the semiring R is said to be a k-ideal in since P is a k-ideal in R, it follows that R if b ∈ B and r ∈ R, then b + r ∈ B (or r + b ∈ B) X X implies r ∈ B. Such ideals were of special interest to S. ( fi)( gj) ∈ P, Bourne [3], M. Henrikson [7], D. R. La Torree [8] and P. i≥i0 j>j0 J. Allen [2]. Theorem 4.10. Let B be a proper, homogeneous ideal a contradiction. Therefore, F,G 6∈ P imply FG 6∈ P , and √ √ it follows that P is a completely prime ideal in R. ut in a graded, R. If B is a k-ideal in R, then B is a Let R be a graded semiring. Suppose that B = P ∩ homogeneous ideal in R. 1 √ Pn · · · ∩ Pn, where the Pi are prime homogeneous k-ideals. Proof. Let p be a non-zero element in B and let rq i=1 i Suppose also that fg ∈ B, fi 6∈ Pi implies g ∈ Pi for i = be the decomposition of p into its homogeneous√ compo- 1, ··· , n when f and g are homogeneous elements. Then nents, where q1 < q2 < ··· < qn. Since p ∈ B, there it follows that FG ∈ B implies, for each i = 1, ··· , n, if exists a positive integer m such that pm ∈ B. Thus, √ F√ 6∈ P√i then G ∈ Pi. Furthermore, B = B and B = C ∩ D = C ∩ D, where C = ∩{P |F ∈ P }, D = Xn i i r m + X = ( r )m = pm ∈ B, {Pj|G ∈ Pj}. Here if ∩{Pi|F ∈ Pi} contains no Pi, we q1 qi let C = R, while if ∩{P |G ∈ P } contains no P , we let i=1 i j j D = R. where X consists of terms of degree greater that mq1. m m Thus, rq1 is the homogeneous component of p of degree mq . Since B is homogeneous, it is clear that r m ∈ 5. Conclusion 1 √ √ √ q1 B. Thus, rq1 ∈ B. Since p, rq1 ∈ B and B is a k- Pn √ Pn In this paper we have considered the situation where R is ideal in R, p = rq1 + i=2 rqi ∈ B implies i=2 rqi ∈ √ √ Pn an A-semiring and where the semiring is graded so that B. By the same argument, rq ∈ B and rq ∈ √ 2 i=3 i one may√ consider proper homogeneous ideals B and their B. Continuing in this fashion, it is√ clear that the homo- radicals B which were shown to be homogeneous as geneous components of p belong to B. ut well if they were k-ideals and furthermore necessary and To conclude this paper, necessary and sufficient condi- sufficient conditions were found for a homogeneous k- tions will be given in order that a homogeneous k-ideal in ideal P to be completely prime. Looking forward ques- a graded semiring be completely prime. tions arise concerning the softening of conditions under Theorem 4.11. Let P be a homogeneous k-ideal in the which the equivalents (i.e., generalizations) of Theorems graded semiring R. In order that P be completely prime in 4.10 and 4.11 may be deduced. Thus, one may look at the R, it is necessary and sufficient that fg ∈ P implies f ∈ P role played by A-semirings in the arguments above and or g ∈ P for homogeneous elements in R. of course the question about the condition on ideals be- P Proof. PLet F and G be elements in R − P and let ing k-ideals as being replaceable through conditions on fi and gj be the decompositions of F and G, re- the semiring R itself. The extents to which all this may spectively, into their homogeneous components. Clearly, be done is a set of questions and topics of possible interest F 6∈ P and G 6∈ P imply there exist homogeneous com- in future investigations. ponents of least degree fi0 and gj0 of F and G, respec- P f g 6∈ P tively, whichP are notP in . Consequently, i0 j0 . As- sume ( fi)( gj) ∈ P . Thus, fi gj + X ∈ i≥i0 j≥j0 0 0 References P where X consists of terms of degree greater that i0 + j0. Since fi0 gj0 is the homogeneous component of de- [1] P. J. Allen, H. S. Kim and J. Neggers, Kyungpook Math. J. gree i0 + j0 of an element in the homogeneous ideal P , (to appear). f g ∈ P itP is clear thatP i0 j0 , a contradiction. Therefore, [2] P. J. Allen and R. Windham, Publications Mathematicae ( f )( g ) 6∈ P . Assume FG ∈ P . Thus, the (Debrecen) 20 (1973), 161. i≥i0 i j≥j0 j

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[3] S. Bourne, Proc. Nat. Acad. Sci., U. S. A. 38 (1952), 118. Joseph Neggers received a Ph.D. [4] J. S. Golan, Semirings and their applications, Kluwer Aca- from the Florida State University demic, Boston, 1999. in 1963. After positions at the Florida [5] J. S. Han, H. S. Kim and J. Neggers, Appl. Math. & Inf. Sci. State University, the University of 6 (2012), 99. Amsterdam, King‘s College (Lon- [6] U. Hebisch and H. J. Weinert, Semirings, algebraic theory don, UK), and the University of and applications in computer science, World Scientiﬁc, Sin- Puerto Rico, he joined the Univer- gapore, 1993. sity of Alabama in 1967, where [7] M. Henrikson, Amer. Math. Soc. Notices 6 (1958), 321. he is still engaged in teaching, re- [8] D. R. La Torree, Publication Mathematicae (Debrecen) 12 search and writing poetry often in (1965), 219. a calligraphic manner as well as enjoying friends and fam- ily through a variety of media both at home and abroad. He has reviewed over 500 papers in Zentralblatt Math., P. Allen is a professor of the and published over 80 research papers. In addition he has Department of Mathematics, Uni- published a book, Basic Posets, with Professor Kim. He versity of Alabama since 1967. He has done research in several areas including algebra, poset has received his Ph.D. at Texas Chris- theory, algebraic graph theory and combinatorics and in- tian University. He has served aca- cluding topics which are of an applied as well as a pure demic advising committee and also nature. dissertation director for many Ph.D. students. He studied several areas of algebras including the theory of semirings, group rings, d/BCK- algebras. He also has interest in mathematics education.

Hee Sik Kim is working at Dept. of Mathematics, Hanyang Univer- sity as a professor. He has received his Ph.D. at Yonsei University. He has publised a book, Basic Posets with professor J. Neggers, and published 150 papers in several journals. He is working as an (man- aging) editor of 3 journals. His mathematical research areas are BCK- algebras, fuzzy algebras, poset theory and theory of semirings, and he is reviewing many papers in this areas. He is engaged in martial arts, photography and poetry also.

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