Almost weakly finite conductor rings and weakly finite conductor rings
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Let $R$ be a commutative ring with identity. We call the ring $R$ to be an almost weakly finite conductor if for any two elements $a$ and $b$ in $R$, there exists a positive integer $n$ such that $a^{n}R\cap b^{n}R$ is finitely generated. In this article, we give some conditions for the trivial ring extensions and the amalgamated algebras to be almost weakly finite conductor rings. We investigate the transfer of these properties to trivial ring extensions and amalgamation of rings. Our results generate examples which enrich the current literature with new families of examples of non-finite conductor weakly finite conductor rings.
Abstract
Let $R$ be a commutative ring with identity. We call the ring $R$ to be an almost weakly finite conductor if for any two elements $a$ and $b$ in $R$, there exists a positive integer $n$ such that $a^{n}R\cap b^{n}R$ is finitely generated. In this article, we give some conditions for the trivial ring extensions and the amalgamated algebras to be almost weakly finite conductor rings. We investigate the transfer of these properties to trivial ring extensions and amalgamation of rings. Our results generate examples which enrich the current literature with new families of examples of non-finite conductor weakly finite conductor rings.
참고문헌 (21)
- Khalid Adarbeh, "Matlis’ semi-regularity in trivial ring extensions of integral domains", Colloquium Mathematicum, 150(2), pp.229~241 (2017), Institute of Mathematics, Polish Academy of Sciences[DOI]
- D. D. Anderson, "Idealization of a Module", Journal of Commutative Algebra, 1(1), pp.3~56 (2009), Rocky Mountain Mathematics Consortium[DOI]
- C. Bakkari, "Trivial extensions defined by Prüfer conditions", Journal of Pure and Applied Algebra, 214(1), pp.53~60 (2010), Elsevier BV[DOI]
- Chahrazade Bakkari, "On weakly coherent rings", Rocky Mountain Journal of Mathematics, 44(3), pp.743~752 (2014), Rocky Mountain Mathematics Consortium[DOI]
- S. U. Chase, "Direct products of modules", Trans. Amer. Math. Soc, 97, pp.457~473 (1960)[DOI]
- Robert Damiano, "Commutative torsion stable rings", Journal of Pure and Applied Algebra, 32(1), pp.21~32 (1984), Elsevier BV[DOI]
- M. D'Anna, "Commutative Algebra and Its Applications", pp.155~172 (2009), Walter de Gruyter
- Marco D’Anna, "Properties of chains of prime ideals in an amalgamated algebra along an ideal", Journal of Pure and Applied Algebra, 214(9), pp.1633~1641 (2010), Elsevier BV[DOI]
- Marco D'Anna, "New Algebraic Properties of an Amalgamated Algebra Along an Ideal", Communications in Algebra, 44(5), pp.1836~1851 (2016), Informa UK Limited[DOI]
- Marco D’Anna, "The amalgamated duplication of a ring along a multiplicative-canonical ideal", Arkiv för Matematik, 45(2), pp.241~252 (2007), International Press of Boston[DOI]
- MARCO D'ANNA, "AN AMALGAMATED DUPLICATION OF A RING ALONG AN IDEAL: THE BASIC PROPERTIES", Journal of Algebra and Its Applications, 06(03), pp.443~459 (2007), World Scientific Pub Co Pte Lt[DOI]
- D. E. Dobbs, "On going down for simple overrings", Proc. Amer. Math. Soc, 39, pp.515~519 (1973)[DOI]
- H. El Alaoui, "Homolog-ical and combinatorial methods in algebra", pp.117~126 (2018), Springer
- S. Glaz, "Commutative Coherent Rings" (1989), Springer-Verlag
- Sarah Glaz, "Finite conductor rings", Proceedings of the American Mathematical Society, 129(10), pp.2833~2843 (2000), American Mathematical Society (AMS)[DOI]
- J. A. Huckaba, "Commutative rings with zero divisors" (1988), Marcel Dekker, Inc
- Karima Alaoui Ismaili, "On (n,d)-property in amalgamated algebra", Asian-European Journal of Mathematics, 09(01), pp.1650014~ (2016), World Scientific Pub Co Pte Ltd[DOI]
- S. Kabbaj, "Matlis semi-regularity and semi-coherence in trivial ring extensions: a survey", Moroccan Journal of Algebra and Geometry with Applications
- Najib Mahdou, "On Weakly Finite Conductor Rings", Communications in Algebra, 32(10), pp.4027~4036 (2004), Informa UK Limited[DOI]
- M. Nagata, "Local Rings" (1962), Interscience Publishers a division of John Wiley & Sons
- Muhammad Zafrullah, "A general theory of almost factoriality", Manuscripta Mathematica, 51(1-3), pp.29~62 (1985), Springer Science and Business Media LLC[DOI]