Strongly homogeneous primary ideals in a graded ring

summary:Let $R=\bigoplus _{\alpha \in \Gamma } R_\alpha $ be a commutative ring graded by an arbitrary torsionless grading monoid $\Gamma $. We call a graded primary ideal $P$ of $R$ to be strongly homogeneous primary if $a P \subseteq b R$ or $b^n R \subseteq a^n P$ for some positive integer $n$, for every homogeneous elements $a$, $b$ of $R$. The paper examines the concept of strongly homogeneous primary in graded rings, aiming to deepen the understanding of strongly primary ideals within the ungraded contexts. It examines the essential properties of these ideals, highlighting how they differ from their ungraded counterparts and establishing a relationship with strongly homogeneous prime ideals. The study also explores these graded ideals in particular types of graded rings, such as graded trivial ring extensions and graded amalgamated duplications.