Revista de la
Unión Matemática Argentina
Graded almost valuation rings
Fatima Zahra Guissi, Najib Mahdou, Ünsal Tekir, and Suat Koç

Volume 68, no. 2 (2025), pp. 535–553    

Published online (final version): October 8, 2025

https://doi.org/10.33044/revuma.4036

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Abstract

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a commutative ring graded by an arbitrary torsionless monoid $\Gamma$. We say that $R$ is a graded almost valuation ring (gr AV- ring) if for every two homogeneous elements $a,b$ of $R$, there exists a positive integer $n$ such that either $a^{n}$ divides $b^{n}$ (in $R$) or $b^{n}$ divides $a^{n}$. In this paper, we introduce and study the graded version of the almost valuation ring which is a generalization of gr-AVD to the context of arbitrary $\Gamma$-graded rings (with zero- divisors). Next, we study the possible transfer of this property to the graded trivial ring extension $A\ltimes E$. Our aim is to provide examples of new classes of $\Gamma$-graded rings satisfying the above mentioned property.

References

  1. M. M. Ali, Idealization and theorems of D. D. Anderson. II, Comm. Algebra 35 no. 9 (2007), 2767–2792.  DOI  MR  Zbl
  2. D. D. Anderson, K. R. Knopp, and R. L. Lewin, Almost Bézout domains. II, J. Algebra 167 no. 3 (1994), 547–556.  DOI  MR  Zbl
  3. D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 no. 1 (2009), 3–56.  DOI  MR  Zbl
  4. D. D. Anderson and M. Zafrullah, Almost Bézout domains, J. Algebra 142 no. 2 (1991), 285–309.  DOI  MR  Zbl
  5. D. D. Anderson and M. Zafrullah, Almost Bézout domains. III, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51(99) no. 1 (2008), 3–9.  MR  Zbl
  6. A. Assarrar, N. Mahdou, Ü. Tekir, and S. Koç, On graded coherent-like properties in trivial ring extensions, Boll. Unione Mat. Ital. 15 no. 3 (2022), 437–449.  DOI  MR  Zbl
  7. A. Badawi, On pseudo-almost valuation domains, Comm. Algebra 35 no. 4 (2007), 1167–1181.  DOI  MR  Zbl
  8. C. Bakkari, N. Mahdou, and A. Riffi, Graded almost valuation domains, Vietnam J. Math. 49 no. 4 (2021), 1141–1150.  DOI  MR  Zbl
  9. J. A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics 117, Marcel Dekker, New York, 1988.  MR  Zbl
  10. R. Jahani-Nezhad and F. Khoshayand, Almost valuation rings, Bull. Iranian Math. Soc. 43 no. 3 (2017), 807–816.  MR  Zbl
  11. N. Jarboui and D. E. Dobbs, On almost valuation ring pairs, J. Algebra Appl. 20 no. 10 (2021), Paper no. 2150182.  DOI  MR  Zbl
  12. C. Jayaram and Ü. Tekir, von Neumann regular modules, Comm. Algebra 46 no. 5 (2018), 2205–2217.  DOI  MR  Zbl
  13. M. Kabbour and N. Mahdou, On valuation rings, Comm. Algebra 39 no. 1 (2011), 176–183.  DOI  MR  Zbl
  14. I. Kaplansky, Commutative rings, revised ed., University of Chicago Press, Chicago, Ill.-London, 1974.  MR  Zbl
  15. N. Mahdou, A. Mimouni, and M. A. S. Moutui, On almost valuation and almost Bézout rings, Comm. Algebra 43 no. 1 (2015), 297–308.  DOI  MR  Zbl
  16. C. Năstăsescu and F. Van Oystaeyen, Methods of graded rings, Lecture Notes in Math. 1836, Springer, Berlin, 2004.  DOI  MR  Zbl
  17. D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, London, 1968.  MR  Zbl
  18. M. Refai, M. Hailat, and S. Obiedat, Graded radicals and graded prime spectra, Far East J. Math. Sci. (FJMS) Special Volume, Part I (2000), 59–73.  MR  Zbl