On 2-N-prime ideals over commutative ℤ2-graded ring

Melhem, Alaa; Abu-Dawwas, Rashid; Albalasmeh, Ali

doi:10.47974/JDMSC-2579

Open Access Research Article

On 2-N-prime ideals over commutative ℤ2-graded ring

Alaa Melhema.melhem@jadara.edu.joDepartment of Mathematics Faculty of Science Jadara UniversityIrbid, JordanView full profile → , ^*Rashid Abu-DawwasCorresponding authorrabudawwas@pmu.edu.saDepartment of Mathematics and Natural Sciences College of Sciences and Human Studies Prince Mohammad Bin Fahd University P. O. Box 1664Al Khobar, 31952, Saudi ArabiaView full profile → , Ali Albalasmeh2020105004@ses.yu.edu.joDepartment of Mathematics Yarmouk UniversityIrbid, 21163, JordanView full profile →

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pp. 1335–1346Vol. 29Issue 3March 2026DOI: 10.47974/JDMSC-2579 Crossmark XML

Received:: 11 Mar 2025
Published Online:: 05 Mar 2026
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2579
Pages:: 1335–1346

Abstract

Let R be a commutative ring with a nonzero unity element. In this article, we introduce and examine new classes of ideals in ℤ2-graded rings, expanding upon the earlier concept of N-prime ideals. Using the function N : R → R0, defined by N(x) = x02 – x12 for x = x0 + x1 ∈ R, we define and analyze two new types of ideals, 2-N-prime and weakly 2-N-prime ideals. A proper ideal I is called 2-N-prime if xy ∈ I implies that N(x2) ∈ I or N(y2) ∈ I. Similarly, I is weakly 2-N-prime if 0 ≠ xy ∈ I implies that N(x2) ∈ I or N(y2) ∈ I. We explore the fundamental properties of these ideals, highlighting their connections to established structures in graded ring theory. Through examples and counter examples, we illustrate the differences between 2-N-prime and weakly 2-N-prime ideals, underscoring their significance in the theory of graded rings. These results not only deepensour understanding of ideal theory in ℤ2-graded rings but also open new avenues for futures research.

Keywords

ℤ2-graded rings Prime ideals 2-prime ideals Weakly 2-prime ideals

Subject Classifications

13A0213A15

References

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