Flat semimodules

To my dearest friend Najla Ali We introduce and investigate flat semimodules and k-flat semimodules. We hope these concepts will have the same importance in semimodule theory as in the theory of rings and modules. 1. Introduction. We introduce the notion of flat and k-flat. In Section 2, we study the structure ensuing from these notions. Proposition 2.4 asserts that V is flat if and only if (V ⊗ R −) preserves the exactness of all right-regular short exact sequences. Proposition 2.5 gives necessary and sufficient conditions for a projective semimodule to be k-flat. In Section 3, Proposition 3.3 gives the relation between flatness and in-jectivity. In Section 4, Proposition 4.1 characterizes the k-flat cancellable semimodules with the left ideals. Proposition 4.4 describes the relationship between the notions of projectivity and flatness for a certain restricted class of semirings and semimodules. Throughout, R will denote a semiring with identity 1. All semimodules M will be left R-semimodules, except at cited places, and in all cases are unitary semimodules, that