Essential ideals in semirings

Abstract

The purpose of this paper is to define and study essential ideals in semirings. The entire paper is restricted to hemirings. In Chapter I, the right singular ideal, Z(H), of a hemiring is defined, and the following result is established: if H is a semi-subtractive halfring with commutative multiplication and zero singular ideal, then every non-zero ideal contains a non-zero k-ideal. Also, several results concerning completely reducible hemirings are obtained. In Chapter II, essentially nilpotent ideals are investigated. It is shown that the prime radical of an arbitrary hemiring is essentially nilpotent and, also, that every nil ideal is essentially nilpotent in a hemiring which satisfies the ascending chain condition on principal right annihilators. The following result is established: if H is a nil hemiring, then H is right essentially nilpotent if and only if Z(H) is both right essential in H and right essentially nilpotent. Hemirings which satisfy the descending chain condition on k-essential right ideals are studied in Chapter III. Such hemirings are called min-E hemirings. It is shown that the Jacobson radical of a min-E, semi-subtractive halfring with 1 is nilpotent. A similar result is obtained for several other classes of min-E halfrings. In Chapter IV, right quotient hemirings are defined and investigated. It is shown that if Q is a right quotient bemiring of H and if Q is regular, then Z(H) is zero. Under additional hypotheses, a converse to this result is obtained. The relationship between the right quotient halfring of a halfring H and the right quotient ring of H-bar is studied.