SSEP Modules and Trivial Extensions

In this paper, we study the behavior of modules possessing the Summand Sum Essential Property (SSEP) in the context of trivial extensions of rings. We provide a detailed examination of how the SSEP and the classical Summand Sum Property (SSP) are preserved or characterized when extending a ring by an (R, R)-bimodule. Necessary and sufficient conditions are established under which the trivial extension inherits these properties from the base ring. Our results generalize existing findings on summand properties and contribute to a deeper understanding of module decompositions and essentiality in extended algebraic structures. Several illustrative examples are provided to demonstrate the applicability of the theoretical developments.