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.
In modules there is a relation between supplemented and π-projective semimodules. This relation was introduced, explained and investigated by many authors. This research will firstly introduce a concept of "supplement subsemimodule" analogues to the case in modules: a subsemimodule Y of a semimodule W is said to be supplement of a subsemimodule X if it is minimal with the property X+Y=W. A subsemimodule Y is called a supplement subsemimodule if it is a supplement of some subsemimodule of W. Then, the concept of supplemented semimodule will be defined as follows: an S-semimodule W is said to be supplemented if every subsemimodule of W is a supplement. We also review other types of supplemented semimodules. Previously, the concept of π-projective semimodule was introduced. The main goal of the present study is to explain the relation between the two concepts, supplemented semimodule and π-projective semimodules, and prove these relations by many results.
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