$S$-Prime and $S$-maximal ideals in trivial ring extensions of commutative rings

This paper explores the study of $S$-prime and $S$-maximal ideals in the context of trivial ring extensions $A \ltimes M$. Through counterexamples, we demonstrate that $S$-prime (resp., $S$-maximal) ideals in $A \ltimes M$ are not necessarily homogeneous, and a homogeneous $S$-prime (resp., $S$-maximal) ideal does not necessarily have the form $P \ltimes M$, where $P$ is an $S_0$-prime (resp., $S_0$-maximal) ideal of $A$. Moreover, we characterize the conditions under which an ideal $J$ (not necessarily homogeneous) in the trivial ring extension $A \ltimes M$ is $S$-prime (resp., $S$-maximal). Additionally, we demonstrate that all $S$-prime (and consequently $S$-maximal) ideals in $A \ltimes M$ are of the form $P \ltimes M$, where $P$ is an $S_0$-prime ideal of $A$, if and only if $M$ is an $S_0$-divisible $A$-module. As an application, we explore the transfer of the concepts of compactly $S$-packed rings, coprimely $S$-packed rings and $S$-$pm$-rings to the trivial ring extension. These results provide significant insights into the relation between $S$-primality and $S$-maximality in trivial ring extensions, contributing to a deeper understanding of ideal theory in this context. This work not only enriches the theoretical framework of ring structures but also advances the broader field of algebraic theory through practical examples and applications.