대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제37권2호
  • /
  • Pages.337-345
  • /
  • 2022
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

SOME RESULTS ON S-ACCR PAIRS

  • Hamed, Ahmed (Department of Mathematics Faculty of Sciences) ;
  • Malek, Achraf (Department of Mathematics Faculty of Sciences)
  • 투고 : 2021.04.02
  • 심사 : 2021.07.23
  • 발행 : 2022.04.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let R ⊆ T be an extension of a commutative ring and S ⊆ R a multiplicative subset. We say that (R, T) is an S-accr (a commutative ring R is said to be S-accr if every ascending chain of residuals of the form (I : B) ⊆ (I : B2) ⊆ (I : B3) ⊆ ⋯ is S-stationary, where I is an ideal of R and B is a finitely generated ideal of R) pair if every ring A with R ⊆ A ⊆ T satisfies S-accr. Using this concept, we give an S-version of several different known results.

키워드

참고문헌

  1. H. Ahmed and H. Sana, S-Noetherian rings of the forms A[X] and A[[X]], Comm. Algebra 43 (2015), no. 9, 3848-3856. https://doi.org/10.1080/00927872.2014.924127
  2. H. Ahmed and H. Sana, Modules satisfying the S-Noetherian property and S-ACCR, Comm. Algebra 44 (2016), no. 5, 1941-1951. https://doi.org/10.1080/00927872.2015.1027377
  3. D. D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra 30 (2002), no. 9, 4407-4416. https://doi.org/10.1081/AGB-120013328
  4. N. Bourbaki, Commutative Algebra, Addison-Wesley, Reading. MA. 1972.
  5. D. E. Dobbs, Ahmes expansions of formal Laurent series and a class of non-Archimedean integral domains, J. Algebra 103 (1986), no. 1, 193-201. https://doi.org/10.1016/0021-8693(86)90177-8
  6. A. Hamed and A. Malek, S-prime ideals of a commutative ring, Beitr. Algebra Geom. 61 (2020), no. 3, 533-542. https://doi.org/10.1007/s13366-019-00476-5
  7. C.-P. Lu, Modules and rings satisfying (accr), Proc. Amer. Math. Soc. 117 (1993), no. 1, 5-10. https://doi.org/10.2307/2159690
  8. W. Maaref, A. Benhissi, and A. Hamed, On S-Artinian rings and modules, submitted.
  9. E. S. Sevim, U. Tekir, and S. Koc, S-Artinian rings and finitely S-cogenerated rings, J. Algebra Appl. 19 (2020), no. 3, 2050051, 16 pp. https://doi.org/10.1142/S0219498820500516
  10. P. B. Sheldon, How changing D[[x]] changes its quotient field, Trans. Amer. Math. Soc. 159 (1971), 223-244. https://doi.org/10.2307/1996008
  11. S. Visweswaran, ACCR pairs, J. Pure Appl. Algebra 81 (1992), no. 3, 313-334. https://doi.org/10.1016/0022-4049(92)90063-L