ON S-ENDO-NOETHERIAN RINGS
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A commutative ring A is said to be an S-endo-Noetherian ring, where S is a multiplicative subset of A, if the chain of annihilators ann_A(a₁) ⊆ ann_A(a₂) ⊆ ⋯ is S-stationary; that is, there exist s in S and a positive integer n such that s·(ann_A(a_k)) ⊆ ann_A(a_n) for each k ≥ n for each sequence (a_k) of elements of A. We study several properties of an S-endo-Noetherian ring. Among other results, the transfer of this property to the amalgamated duplication of A along an ideal I (A ⋈ I) is investigated, as well as the necessary and sufficient conditions for R[[X]] (respectively, R[X]) to be an S-endo-Noetherian ring.
Abstract
A commutative ring A is said to be an S-endo-Noetherian ring, where S is a multiplicative subset of A, if the chain of annihilators ann_A(a₁) ⊆ ann_A(a₂) ⊆ ⋯ is S-stationary; that is, there exist s in S and a positive integer n such that s·(ann_A(a_k)) ⊆ ann_A(a_n) for each k ≥ n for each sequence (a_k) of elements of A. We study several properties of an S-endo-Noetherian ring. Among other results, the transfer of this property to the amalgamated duplication of A along an ideal I (A ⋈ I) is investigated, as well as the necessary and sufficient conditions for R[[X]] (respectively, R[X]) to be an S-endo-Noetherian ring.
참고문헌 (21)
- D. D. Anderson, "S-Notherian rings", Comm. Algebra, 30(9), pp.4407~4419 (2002)
- D. D. Anderson, "Idealization of a module", J. Commut. Algebra, 1(1), pp.3~56 (2009)
- J. Baeck, "S-Noetherian rings and their extension", Taiwaness J. Math, 20(6), pp.1231~1250 (2016)
- A. Benhissi, "Chain Conditions in Commutative Rings" (2022), Springer Nature
- A. Benhissi, "Chain condition on annihilators and strongly Hopfian property in Hur-witz series ring", Algebra Colloq, 21(4), pp.635~646 (2014)
- M. B. Boisen, "CPI-extension : Over rings of integral domains with special prime spectrum", Canad. J. Math, 29(4), pp.722~737 (1977)
- M. D'Anna, "Amalgamated algebras along an ideal", Comm. Algebra and Applications, Walter de Gruyter, Berlin, pp.155~172 (2009)
- M. D'Anna, "A construction of Gorenstein rings", J. Algebra, 306(2), pp.507~519 (2006)
- M. D'ana, "The amalgamated duplication of a ring along a multiplication-canonical ideal", Ark. Mat, 45(2), pp.241~252 (2007)
- M. D'Anna, "An amalgamated duplication of a ring along an ideal : the basic properties", J. Algebra Appl, 6(3), pp.443~459 (2007)
- A. El Khalfi, "Amalgamation extension in commutative ring theory, a survey", Moroccan Journal of Algebra and Geometry with Applications, 1(1), pp.139~182 (2022)
- B. Gouaid, "Endo-Noetherian rings", Ann. Mat. Pur. Appl, 199(2), pp.563~572 (2020)
- A. Hamed, "Modules Satisfying the S-Noetherian Property and S-ACCR", Comm. Algebra, 44(5), pp.1941~1951 (2006)
- "Comm. Algebra", 2015, 3848~3856
- A. Hamed, "Modules satisfying the S-Noetherian property and S-ACCR", Comm. Algebra, 44(5), pp.1941~1951 (2016)
- N. K. Kim, "Armendariz ring and reduced ring", J. Algbra, 223(2), pp.477~488 (2000)
- J. W. Lim, "A note on S-Noetherian domains", Kyungpook Math. J, 55(3), pp.507~514 (2015)
- J. W. Lim, "S-Noetherian properties on amalgamated algebras along an ideal", J. Pure Appl. Algebra, 218(6), pp.1075~1080 (2014)
- M. Nagata, "Local Rings" (1962), Wiley-Interscience
- M. A. Ndiaye, "On commutative EKFN-ring with ascending chain condition on annihilators", Int. J. App. Math, 86(5), pp.871~881 (2013)
- E. Sengelen Sevim, "S-Artinian rings and finitely S-cogenerated rings", J. Algebra Appl., 19(3), pp.2050051~ (2020)