Extending Semimodules over Semirings

Abstract

The objective of our research paper is to introduce as well as to study many essential properties of the concept of extending semimodules. A semimodule S is named extending (CS) if every subsemimodule of S is essential in a direct summand of S. Therefore, extending semimodule behaviour with respect to direct sums and direct summands are examined. Moreover, studying some properties of these semimodules concepts, e.g., every direct summand of a CS-semimodule is a CS-semimodule. While the direct sum of extending semimodules is not necessarily extending.

Almost Injective Semimodules

“…It is clear that K is closed subsemimodule if and only if K is a complement in ℳ [6]. An R-semimodule ℳ is called CS-semimodule if every subsemimodule of ℳ is large in direct summand of ℳ, equivalently, every closed subsemimodule of ℳ is direct summand of it [12]. Let ℳ be an R-semimodule and L be a subsemimodule of ℳ, then ℳ is said to be maximal essential extension of L if 𝒩 is proper extension of ℳ, then 𝒩 is not essential extension of L [6].…”