Structural Characterizations of Primeness in Semigroups

This study investigates the set $P_{S} = \{ b \in S \mid SSb = (0) \}$ for a semigroup $S$ with zero, where $P_{S}$ is referred to as the \emph{source of primeness} of $S$. The paper first explores the fundamental algebraic properties of $P_{S}$, establishing its structural role within the semigroup. It then provides a characterization of $\lvert P_{S}\rvert$-regular, $\lvert P_{S}\rvert$-idempotent, $\lvert P_{S}\rvert$-reduced, and $\lvert P_{S}\rvert$-nonzero-divisor semigroups. Furthermore, the work considers the relationship between the notions of semiprimeness and primeness in semigroups, highlighting how the source of primeness can serve as a tool to analyze and distinguish these concepts. An illustrative example is presented to demonstrate the applicability of the results.