Left Goldie semirings

Abstract

The purpose of this paper is to define and study left Goldie semirings. The entire paper is restricted to hemirings. In Chapter I, the prime radical P(H) of a hemiring is shown to beak-ideal. P(H) is characterized as the set of all strongly nilpotent elements of H. The relationship between the prime radical of a halfring and the prime radical of the ring of differences is studied. Also, the following result is established: if H is a semi-prime semi-subtractive halfring, then every non-zero left (right) ideal contains a non-zero left (right) k-ideal. In Chapter II, left Goldie semirings are defined and nilpotency of suitably conditioned nil structures is investigated. In particular, it is shown that every nil subsemiring of a semi-subtractive left Goldie halfring is nilpotent. The concept of a quotient semiring of a semiring by a sub-semigroup of the semiring is outlined in Chapter III. Various relations between the semiring and its quotient semiring are established. In Chapter IV, essential ideals are introduced and studied. The left singular ideal is also defined and investigated. It is shown that a semi-prime semi-subtractive left Goldie halfring is a left order in a semi-prime semi-subtractive halfring which satisfies both the ascending and descending chain conditions on all left k-ideals. Finally, in Chapter V, the relation between a halfring and its ring of differences is considered. It is shown that a semi-subtractive halfring is left Goldie if and only it its ring of differences is left Goldie. The total left quotient semiring of the ring of differences of H is the ring of differences of the total left quotient semiring of H.