Nil and Nilpotent Ideals over Hemirings

In this paper the notion of subtractive ideal and completely subtractive he miring are introduced. In the third section it is proved that the definition of nil ideal over an art in he miring is in fact equivalent to the definition of nilpotent ideal. Moreover, we show that an art in (noether) he miring R which is completely subtractive has a unique maximal nilpotent ideal B satisfying that B contains all nilpotent one-sided ideals of R and the factor he miring R/B doesn't have nilpotent ideals.