Toward Homological Characterization of Semirings by e-Injective   Semimodules
  In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective. 
J. Y. Abuhlail, S. N. Il'in, Y. Katsov and T. G. Nam
math.RA