  In this paper first by the fact that the relation $\alpha^*$ is the
transitive closure of two its subrelations we introduce and analyze a binary
relation $\lambda^*_e$ on a hyperring such that the derived ring is a unitary
ring. Next we introduce and study the notion of $\lambda_e$-parts in a
hyperring and we characterize $\lambda_e$-parts in a $\lambda_e$-strong
hyperring $R$. Finally we introduce a new relation $\Lambda^*$ such that its
derived ring be a unitary commutative ring.
