Characterizations of hemirings by their hh-ideals

In this paper we characterize hemirings in which all hh-ideals or all fuzzy hh-ideals are idempotent. It is proved, among other results, that every hh-ideal of a hemiring RR is idempotent if and only if the lattice of fuzzy hh-ideals of RR is distributive under the sum and hh-intrinsic product of fuzzy hh-ideals or, equivalently, if and only if each fuzzy hh-ideal of RR is intersection of those prime fuzzy hh-ideals of RR which contain it. We also define two types of prime fuzzy hh-ideals of RR and prove that, a non-constant hh-ideal of RR is prime in the second sense if and only if each of its proper level set is a prime hh-ideal of RR.