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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:p 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 homogeneousp ideals and their radicalsp 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 62 P; G 62 P implies FG 62 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 opera- tions 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 2 R such that x+0 = x and x0 = 0x = 0 as ideals of a ring, as positive cones of partially ordered for each x 2 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 2 S and xy 2 S tional theoretical mathematical background. The semiring whenever x; y 2 S and (2) 0 2 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 2 I and r 2 R implies with zero and commutative addition, appeared in a natu- a + b 2 I; ra 2 I and ar 2 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 el- ement 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] °c 2013 NSP Natural Sciences Publishing Cor. 88 P. J. Allen et. al. : Ideal theory in graded ... P k ni ni if f(X) = i=1 ani X 2 B implies ani X 2 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 = q2Z 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 fRqgq2Z 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 fRqgq2Z 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 fO(X)g. It (2)If rq1 ; rq2 ; ¢ ¢ ¢ ; rqk are non-zero elements in R where Pk is clear that fR g is a collection of additive subsemi- q 6= q if i 6= j, and each r 2 R , then r is q q2Z 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- q2Z 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] = q2Z Rq collection fR g . Although the collection fR g in q q2Z q q2Z is a graded semiring, where the collection fRqgq2Z 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 fRqgq2Z q q0 direct sum of the collection fRqgq2Z . 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 q2Z 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 ½ R 0 for each q; q 2 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 2 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 fRqgq2Z 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 = q2Z Rq; 0 tion of the notion of a polynomial semiring. (2)RqRq0 ½ Rq+q0 for each q; q 2 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 = f0g if q is a non-zero element in Z. It is clear that q 6= q if i 6= j, and each r 2 R , then r is q i j qi qi i=1 qi fR g is a collection of additive subsemigroups of the a non-zero element in R. q q2Z P semiring R. An inspection shows that R = q2Z Rq is a P Whenever R is a graded semiring, the notation “R = graded semiring. If R has only a finiteP number of elements, q2Z Rq” will be used to indicate that fRqgq2Z is the then the finite, graded semiring R = q2Z 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 = q2Z Rq is a graded semiring, and if n and m are distinct integers, then Rn \ Rm = ; or Rn \ Rm = f0g, where 0 denotes the zero in R.
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