Abstract
This paper studies the transfer of pseudo-almost valuation property (PAVR property for short) to various context of commutative ring extensions such as power series ring, trivial ring extension and amalgamation. Our work is motivated by an attempt to generate new original classes of rings satisfying this property. The obtained results are backed with new and illustrative examples arising as trivial ring extensions, amalgamations and pullback constructions.
References
[1] D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3–56. 10.1216/JCA-2009-1-1-3Search in Google Scholar
[2] N. O. Azaiez and M. A. S. Moutui, Almost valuation property in bi-amalgamations and pairs of rings, J. Algebra Appl. 18 (2019), no. 6, Article ID 1950104. 10.1142/S0219498819501044Search in Google Scholar
[3] N. O. Azaiez and M. A. S. Moutui, Some commutative ring extensions defined by almost Bézout condition, Hacet. J. Math. Stat. 49 (2020), no. 1, 371–379. 10.15672/hujms.552224Search in Google Scholar
[4] A. Badawi, On pseudo-almost valuation domains, Comm. Algebra 35 (2007), no. 4, 1167–1181. 10.1080/00927870601141951Search in Google Scholar
[5] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extensions defined by Prüfer conditions, J. Pure Appl. Algebra 214 (2010), no. 1, 53–60. 10.1016/j.jpaa.2009.04.011Search in Google Scholar
[6] S. Bazzoni and S. Glaz, Gaussian properties of total rings of quotients, J. Algebra 310 (2007), no. 1, 180–193. 10.1016/j.jalgebra.2007.01.004Search in Google Scholar
[7] M. B. Boisen, Jr. and P. B. Sheldon, CPI-extensions: Overrings of integral domains with special prime spectrums, Canadian J. Math. 29 (1977), no. 4, 722–737. 10.4153/CJM-1977-076-6Search in Google Scholar
[8] G. W. Chang, H. Nam and J. Park, Strongly primary ideals, Arithmetical Properties of Commutative Rings and Monoids, Lect. Notes Pure Appl. Math. 241, Chapman & Hall/CRC, Boca Raton (2005), 378–388. 10.1201/9781420028249.ch26Search in Google Scholar
[9] M. D’Anna, C. A. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, Commutative Algebra and its Applications, Walter de Gruyter, Berlin (2009), 155–172. 10.1515/9783110213188.155Search in Google Scholar
[10] M. D’Anna, C. A. Finocchiaro and M. Fontana, Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (2010), no. 9, 1633–1641. 10.1016/j.jpaa.2009.12.008Search in Google Scholar
[11] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2007), no. 2, 241–252. 10.1007/s11512-006-0038-1Search in Google Scholar
[12] L. Fuchs, Über die Ideale arithmetischer Ringe, Comment. Math. Helv. 23 (1949), 334–341. 10.1007/BF02565607Search in Google Scholar
[13] S. Glaz, Commutative Coherent Rings, Lecture Notes in Math. 1371, Springer, Berlin, 1989. 10.1007/BFb0084570Search in Google Scholar
[14] J. A. Huckaba, Commutative Rings with Zero Divisors, Monogr. Textb. Pure Appl. Math. 117, Marcel Dekker, New York, 1988. Search in Google Scholar
[15] R. Jahani-Nezhad and F. Khoshayand, Pseudo-almost valuation rings, Bull. Iranian Math. Soc. 41 (2015), no. 4, 815–824. Search in Google Scholar
[16] C. Jayaram and Ü. Tekir, Von Neumann regular modules, Comm. Algebra 46 (2018), no. 5, 2205–2217. 10.1080/00927872.2017.1372460Search in Google Scholar
[17] S. Kabbaj, K. Louartiti and M. Tamekkante, Bi-amalgamated algebras along ideals, J. Commut. Algebra 9 (2017), no. 1, 65–87. 10.1216/JCA-2017-9-1-65Search in Google Scholar
[18] S.-E. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra 32 (2004), no. 10, 3937–3953. 10.1081/AGB-200027791Search in Google Scholar
[19] I. Kaplansky, Commutative Rings, University of Chicago, Chicago, 1974. 10.1007/BFb0068926Search in Google Scholar
[20] N. Mahdou, A. Mimouni and M. A. S. Moutui, On almost valuation and almost Bézout rings, Comm. Algebra 43 (2015), no. 1, 297–308. 10.1080/00927872.2014.897586Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- B-essential spectra of 2 × 2 block operator matrix pencils
- New results on perturbations of p-adic linear operators
- Approximation by matrix transform means with respect to the character system of the group of 2-adic integers
- Pentactions and action representability in the category of reduced groups with action
- On interpolation of reflexive variable Lebesgue spaces on which the Hardy–Littlewood maximal operator is bounded
- Further inequalities for the 𝔸-numerical radius of certain 2 × 2 operator matrices
- A generalization of practical stability of nonlinear impulsive systems
- Properties of rational Fourier series and generalized Wiener class
- Some identities on degenerate hyperharmonic numbers
- Generalized Pohozhaev’s identity for radial solutions of p-Laplace equations
- Classes of trivial ring extensions and amalgamations subject to pseudo-almost valuation condition
- Anisotropic nonlinear weighted elliptic equations with variable exponents
- Finite groups in which every non-nilpotent subgroup is a TI-subgroup or has p;'-order
- Finite-time attractivity of strong solutions for generalized nonlinear abstract Rayleigh–Stokes equations
- Reproducing kernels and minimal solutions of elliptic equations
Articles in the same Issue
- Frontmatter
- B-essential spectra of 2 × 2 block operator matrix pencils
- New results on perturbations of p-adic linear operators
- Approximation by matrix transform means with respect to the character system of the group of 2-adic integers
- Pentactions and action representability in the category of reduced groups with action
- On interpolation of reflexive variable Lebesgue spaces on which the Hardy–Littlewood maximal operator is bounded
- Further inequalities for the 𝔸-numerical radius of certain 2 × 2 operator matrices
- A generalization of practical stability of nonlinear impulsive systems
- Properties of rational Fourier series and generalized Wiener class
- Some identities on degenerate hyperharmonic numbers
- Generalized Pohozhaev’s identity for radial solutions of p-Laplace equations
- Classes of trivial ring extensions and amalgamations subject to pseudo-almost valuation condition
- Anisotropic nonlinear weighted elliptic equations with variable exponents
- Finite groups in which every non-nilpotent subgroup is a TI-subgroup or has p;'-order
- Finite-time attractivity of strong solutions for generalized nonlinear abstract Rayleigh–Stokes equations
- Reproducing kernels and minimal solutions of elliptic equations