Received June 14, 2025. Published online February 11, 2026.
Abstract: 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.
Affiliations: Nassima Guennach, Najib Mahdou Laboratory of Modelling and Mathematical Structures, Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco, e-mail: nassima.guennach@usmba.ac.ma, mahdou@hotmail.com; Ünsal Tekir (corresponding author), Suat Koç, Department of Mathematics, Marmara University, Yerleşkesi 34722 Kadiköy, Istanbul, Turkey, e-mail: utekir@marmara.edu.tr, suat.koc@marmara.edu.tr
