On graded 1-absorbing \(\delta\)-primary ideals (Q6586851)

This paper introduces and studies the concept of graded 1-absorbing \(\delta\)-primary ideals in commutative graded rings. By extending the notion of 1-absorbing \(\delta\)-primary ideals to graded rings, the authors explore how this new class of ideals interacts with other established classes, such as graded \(\delta\)-primary ideals and graded 1-absorbing prime ideals. The paper further examines the behavior of graded 1-absorbing \(\delta\)-primary ideals in localized graded rings and graded idealization, offering several key results and illustrative examples.\N\NThe paper provides a valuable contribution by expanding the theory of graded rings through the introduction of graded 1-absorbing \(\delta\)-primary ideals. This new class builds upon existing structures in ideal theory and opens up new avenues for research in the algebraic properties of graded rings. Notably, the results in Theorem 2.7 and Proposition 2.4 establish important connections between graded 1-absorbing \(\delta\)-primary ideals and other well-known classes of ideals, offering insights that are both theoretically interesting and potentially impactful for future studies.\N\NThe paper is generally well-organized and presents its definitions, lemmas, and theorems in a logical and coherent manner. The use of examples, such as Example 2.3, helps to clarify the introduced concepts, though the addition of further examples in some sections could enhance the clarity of the more abstract results. The writing is clear, although certain proofs, particularly those in Theorem 2.7 and Proposition 2.4, might benefit from additional explanation to make the reasoning more accessible to a wider audience.\N\NWhile the paper offers a strong contribution, a few areas could be enhanced:\N\begin{itemize}\N\item \textbf{Additional Examples:} Including more examples, particularly in Sections 3 and 4, would help to further illustrate the practical implications of the theoretical results.\N\item \textbf{Clarification in Proofs:} Some of the proofs could be expanded with more detailed intermediate steps to make the arguments easier to follow, especially for readers who may not be deeply familiar with graded ideal theory.\N\item \textbf{Potential Applications:} A brief discussion of potential applications of graded 1-absorbing \(\delta\)-primary ideals in areas such as algebraic geometry or module theory could broaden the appeal of the paper to a wider mathematical audience.\N\end{itemize}\NThis paper makes a valuable contribution to the study of graded rings and ideals, particularly through the introduction of graded 1-absorbing \(\delta\)-primary ideals, which significantly extends existing theory. The results are well-presented and offer important insights into graded algebraic structures. The paper's findings will undoubtedly serve as a solid foundation for further research in this area, making it a valuable addition to the literature on graded ring theory.