Ideal theory in semirings

Abstract

A semiring is a non-empty set on which there are defined two associative binary operations, called addition and multiplication, such that multiplication distributes over addition both from the left and from the right. A non-empty subset I in a semiring R is called an ideal if a+b in I, ar in I and ra in I for each a,b in I and for each r in R. Three aspects of ideal theory are investigated in this paper. The notion of a Q-ideal was defined, 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. Since the theory of ideals plays an important role in the theory of quotient semirings, an intensive study was made of the notions of prime, completely prime, and primary ideals in commutative semirings. The notion of an A-semiring was defined and a characterization of an A-semiring was presented. With the aid of these notions, further algebraic properties of the radical of an ideal in an A-semiring were discovered. It was also shown that a proper Q-ideal I in the semiring R is primary if and only if every zero divisor in R/I is nilpotent. The notion of a graded semiring was defined and investigated. It was shown that every polynomial semiring is a graded semiring. Examples of graded semirings other than polynomial semirings were given. Using the notions of homogeneous and faithful ideals, further algebraic properties of ideals were obtained. Sufficient conditions were given in order that the radical of a homogeneous ideal in a graded, A-semiring be homogeneous. Necessary and sufficient conditions were given in order that a faithful, homogeneous ideal be completely prime in a graded semiring.