Abstract
An injectivity in the category of semimodules over semiring was studied by many authors recently. On the other hand, the concept of injectivity, in the category of modules over ring, was generalized in many different directions. In particular, injective modules relative to preradical were some of those generalizations.
 As an analogue to module theory, in this paper, we introduce and investigate the notion of "injective semimodule relative to Jacobson radical (namely nearly injective semimodule)".
Highlights
Throughout this work, Ş stands for a commutative semiring with identity and a semimodule means a unitary left Ş-semimodule
Let L be a subset of a left Ş-semimodule A L is called subsemimodule of A if L is closed under addition and scalar multiplication
Proof: Assume that both A and B are subsemimodules of the subtractive Şsemimodule W and B is maximal with the property A ∩ B = 0
Summary
Throughout this work, Ş stands for a commutative semiring with identity and a semimodule means a unitary left Ş-semimodule. An Ş-semimodule W is called injective if for every Ş-monomorphism α: A ⟶ B and for each Ş-homomorphism β: A ⟶ W, there is an Ş-homomorphism h: B ⟶ W such that hα = β[1]. Let L be a subset of a left Ş-semimodule A L is called subsemimodule of A if L is closed under addition and scalar multiplication. In this case it is denoted by L ↪ A. Definition 2.14 [1] An Ş-semimodule A is called injective if for every Şmonomorphism α: A ⟶ B and for each Ş-homomorphism β: A ⟶ W, there is an Şhomomorphism h: B ⟶ W such that the following diagram is commutative (i.e. hα = β)
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