Abstract: In this paper, we investigate and characterize the class of A-semirings. A characterization of the Thierrin radical of a proper ideal of an A-semiring is given. More- over, when P is a Q-ideal in the semiring R, it is shown that P is primary if and only if R=P is nilpotent. The concept of semirings was introduced by H. S. Vandiver in 1935 and has since then been studied by many authors (e.g., (1)-(5), (11)-(13)). In several papers from 1956 to 1958, K. Iseki ((7)-(10)) developed a large amount of ideal theory for semirings that are not necessarily commutative under either operation. Many of Iseki's results were topological in nature; however, he gave several characterizations of prime ideals and has defined and studied the Thierrin radical of an ideal. It is the purpose of this paper to present a development of ideal theory for commutative semirings and to connect this theory with the theory developed by Iseki. P. J. Allen ((1)) introduced the notion of a Q-ideal and a construction process was presented by which one can build the quotient structure of a semiring modulo a Q-ideal. Maximal homomorphisms were defined and examples of such homomorphisms were given. Using these notions, the Fundamental Theorem of Homomorphisms for rings was generalized to include a large class of semirings. The results proven in (1) will be used throughout this paper. Since the theory of ideals plays an important role in the theory of quotient semirings, in this paper, we will make an intensive study of the notions of prime, completely prime, and primary ideals in commutative semrings. The notion of an A-semiring will be defined and a characterization of an A-semiring will be presented. With the aid of these notions, further algebraic properties of the radical of an ideal in an A-semiring will be given. It will also be shown that a proper Q-ideal I in the semiring R is primary if and only if every zero divisor in
Publication Year: 2006
Publication Date: 2006-06-01
Language: en
Type: article
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Cited By Count: 19
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