We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.
We investigate left k-Noetherian and left k-Artinian semirings. We characterize such semirings using i-injective semimodules. We prove in particular, a partial version of the celebrated Bass-Papp Theorem for semiring. We illustrate our main results by examples and counter examples.
Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences of semimodules and its relationships with other notions of flatness for semimodules over semirings. We also prove that a subtractive semiring over which every right (left) semimodule is e-flat is a von Neumann regular semiring.
Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the e-injective semimodule, and the i-injective semimodules through several implications, examples and counter examples. Moreover, we provide partial results for the so called Embedding Problem (of semimodules in injective semimodules).
Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over rings are not necessarily equivalent for semimodules over an arbitrary semiring. We study several of these notions, in particular the e-projective semimodules introduced by the first author using his new notion of exact sequences of semimodules. As pushouts of semimodules play an important role in some of our proofs, we investigate them and give a constructive proof of their existence in a way that proved be very helpful.
