Principally 1-Absorbing Right Primary Ideals

Abstract

This paper first defines the 1-absorbing version of principally right primary ideals (P1ARP ideals), generalizing prime ideals, for noncommutative rings. It then investigates various properties of this ideal structure in different ring settings. It obtains some essential results in ring extensions, such as homomorphic images, product rings, local rings, and idealization. While this study enables the obtaining of original results due to structural differences between commutative and noncommutative rings, it shows that some properties valid in commutative rings are preserved. Finally, the paper concludes by discussing two open problems that could guide future studies.

Keywords

• $1$-Absorbing primary ideals
• noncommutative rings
• prime ideals
• pseudo-radicals

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