Abstract
Let A, and N are a semiring ,and a left A- semimodule, respectively. In this work we will discuss two cases: 
 
 The direct summand of π-projective semi module is π-projective, while the direct sum of two π-projective semimodules in general is not π-projective . The details of the proof will be given.
 We will give a condition under which the direct sum of two π-projective semi modules is π-projective, as well as we also set conditions under which π-projective semi modules are projective.
Highlights
Let A, and N are a semiring,and a left A- semimodule, respectively
We will give a condition under which the direct sum of two π-projective semi modules is π-projective, as well as we set conditions under which π-projective semi modules are projective
The direct sum of semimodules, and π-projective semimodules were studied by several authors [1, 3], that means if for every two subsemimodules M and L with M+L=N, there exist f, g ε End (N) such that f+ g=1N, f(N) M and g(N)
Summary
. Assume that N is a π-projective semimodule, there exists a short exact sequence M →N→0, where M is a free semimodule, since N is a π-projective, by assumption M and N are π-projective and so by (i), M N is π-projective, by Proposition (3.2) N is projective
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