Semi-Coessential and Semi-Coclosed Submodules

In this paper, we study and introduce two new concepts in module theory: semi-coessential submodules and semi-coclosed submodules. These notions are generalizations of the classical concepts of coessential and coclosed submodules and aim to provide a broader framework for analyzing module structures. We investigate the main properties of these submodules and examine their relationships with coessential and coclosed submodules. Several characteristics associated with these ideas are explored, including conditions under which a submodule can be considered semi-coessential or semi-coclosed. The definitions are formulated in terms of factor modules and the semi-small condition, allowing a systematic approach to understanding their behavior within an R-module. Various propositions and illustrative examples are provided to demonstrate how these new submodules retain essential features of their classical counterparts while offering increased flexibility for structural analysis. The results highlight that semi-coessential submodules generalize the notion of coessential submodules by relaxing certain constraints, whereas semi-coclosed submodules extend coclosed submodules by providing a more flexible framework for identifying maximal submodules with no proper semi-coessential submodules. Overall, these concepts enrich the study of module theory by offering new tools for examining the internal structure of modules and their submodules, paving the way for further research and potential applications in algebraic structures and related areas. Let U and D be submodules of an R-module F such that U≤D≤F, then U is called  semi-coessential submodule of D in F,  if  D/U ≪_semi  F/U . Moreover, U is said to be semi-coclosed of F, if  D/U ≪_semi  F/U implies that U=D.