On $k$-Weakly Primary Ideals Over Semirings - Atani, Shahabaddin Ebrahimi (Sarajevo Journal of Mathematics, 2024)
Since ideals in rings and semirings are closely related, two-sided $k$-ideals occur frequently in semiring theory. Let $R$ be a commutative semiring. For an ideal of $R$, the notion of $k$-weakly primary ideals is defined. It is shown that this notion inherits most of the essential properties of the weakly primary ideals of a commutative ring (see [1], [4]). For example, it is proved that a $k$-weakly primary ideal $A$ of $R$, that is not primary, satisfies $A^{2} = 0$ and rad ($A$) = rad (0). Also, it is shown that an intersection of a family of $k$-weakly primary ideals, that are not primary, is $k$-weakly primary. 2000 Mathematics Subject Classification. 16Y60