Akademik Veri Yönetim Sistemi
Montes Taurus Journal of Pure and Applied Mathematics, cilt.7, sa.1, ss.137-145, 2025 (Scopus)
In this study, we investigate modules satisfying the property that the intersection of two direct summands is essential in a direct summand, referred to as SIEP modules and similarly, we study modules where the sum of two direct summands is essential in a direct summand, known as SSEP modules. Based on the obtained results, this study enhances the understanding of fundamentality conditions in direct sum decompositions by elucidating the relationships between SIEP and SSEP modules and the previously investigated SIP and SSP modules. The findings presented herein contribute to the broader framework of module theory and ring theory, providing new insights for future research on the structural properties of direct sums and their fundamental components. Subsequently, we discuss the fundamental properties of SSEP and SIEP modules. Furthermore, we examine the structural properties of direct summands of SSEP (SIEP) modules and explore SSEP (SIEP) matrix rings. Finally we prove that K be a ring with identity 1, m a positive integer and R be the ring Matm(K) of all m × m matrices with entries in K, and the h11 denote the matrix in R with (1, 1) entry 1 and all other entries 0, then the R is a right SIEP (SSEP) ring if and only if the free right K-module Km is an SIEP (SSEP) module.