ON THE DIRECT SUM OF GENERALIZED MODULES AND ITS PROPERTIES

Abstract

The development of generalized algebraic structure theory currently focuses on extending the concepts of classical module theory over rings with unity. However, studies regarding the construction and elementary properties of the external direct sum of generalized modules remain underexplored. This research aims to examine the structure of the external direct sum of generalized modules along with the existence of its homomorphic mappings. The method employed is the axiomatic deductive method by analyzing the governing axioms in generalized module and group theories. The results demonstrate that the external direct sum of generalized modules inherently satisfies the axioms of generalized modules and allows for the construction of corresponding submodule structures. Furthermore, this study successfully constructs new morphic mappings via canonical projections and induced mappings from component homomorphisms, which are proven to be generalized module epimorphic. These findings strengthen the structural foundation of generalized module theory and offer potential avenues for further development in generalized factor module constructions.