Semisimple semirings

Abstract

It is the purpose of this paper to extend the theory of Artinian rings and semisimple rings to certain classes of semirings. The entire paper is restricted to hemirings. Various relationships are established between semi-subtractive hemirings, halfrings, and hemirings whose zeroid is zero. In Chapter I, the following result is established: if H is a semi-subtractive inverse free halfring, then every non-zero left (right) ideal contains a non-zero left (right) k-ideal. Also, semi-subtractive halfrings with non-trivial multiplication having only two left (right) k-ideals are shown to be division halfrings. In Chapter II, much of the theory of Artinian rings is extended to Artinian hemirings. W, the sum of all the nilpotent ideals is shown to beak-ideal, and W is nilpotent if H is left (right) Noetherian or Artinian. It is established that the Levitzki radical, L(H), of a hemiring H is a k-ideal. L(H) is characterized as the intersection of all the prime k-ideals P of H such that L(H/P) = (0). Also, every locally nilpotent simple hemiring is shown to be nilpotent. In Chapter III, every left Artinian hemiring such that H* intersection I not equal to (0) for every left k-ideal of H is shown to be a ring. Semisimple semi-subtractive halfrings are characterized in the following way: a hemiring H is a semisimple semi-subtractive halfring if and only if H is a finite direct sum of matrix rings with entries from division rings and at most one division halfring. An example is given to show that the result fails to hold if H is not semi-subtractive. In Chapter IV, two classes of semisimple halfrings are investigated. These classes are obtained by generalizing the concept of semi-subtractivity and placing additional requirements on H. Just as in the case of semisimple semisubtractive halfrings, these two classes are characterized in terms of matrices.