Injective semimodules-revisited

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no nonzero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so-called [Formula: see text]-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 [Formula: see text]-injective semimodule, and the [Formula: see text]-injective semimodules through several implications, examples and counter examples. Moreover, we show that every semimodule over an arbitrary semiring can be embedded in a [Formula: see text]-[Formula: see text]-injective semimodule.

The Jacobson Radical of the Endomorphism Semiring of P.Q.- Principal Injective Semimodules

In this work, we introduced the Jacobson radical (shortly Rad (Ș)) of the endomorphism semiring Ș = ( ), provided that is principal P.Q.- injective semimodule and some related concepts, we studied some properties and added conditions that we needed. The most prominent result is obtained in section three -If is a principal self-generator semimodule, then (ȘȘ) = W(Ș). Subject Classification: 16y60

Principally Quasi-Injective Semimodules

In this work, the notion of principally quasi- injective semimodule is introduced, discussing the conditions needed to get properties and characterizations similar or related to the case in modules. Let be an -semimodule with endomorphism semiring Ș. The semimodule is called principally quasi-injective, if every -homomorphism from any cyclic subsemimodule of to can be extended to an endomorphism of .

Nearly Injective Semimodules

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)".