Abstract
This paper aims at the study of the notions of periodic, UU and semiclean properties in various context of commutative rings such as trivial ring extensions, amalgamations and pullbacks. The results obtained provide new original classes of rings subject to various ring theoretic properties.
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References
D. D. Anderson, N. Bisht: A generalization of semiclean rings. Commun. Algebra 48 (2020), 2127–2142.
N. Arora, S. Kundu: Semiclean rings and rings of continuous functions. J. Commut. Algebra 6 (2014), 1–16.
A. Badawi: On abelian π-regular rings. Commun. Algebra 25 (1997), 1009–1021.
A. Badawi, A. Y. M. Chin, H. V. Chen: On rings with near idempotent elements. Int. J. Pure Appl. Math. 1 (2002), 255–261.
C. Bakkari, M. Es-Saidi: Nil-clean property in amalgamated algebras along an ideal. Ann. Univ. Ferrara Sez. VII Sci. Mat. 65 (2019), 15–20.
H. E. Bell: On quasi-periodic rings. Arch. Math. 36 (1981), 502–509.
M. Chacron: On a theorem of Herstein. Can. J. Math. 21 (1969), 1348–1353.
M. Chhiti, N. Mahdou, M. Tamekkante: Clean property in amalgamated algebras along an ideal. Hacet. J. Math. Stat. 44 (2015), 41–49.
M. D’Anna, C. A. Finocchiaro, M. Fontana: Amalgamated algebras along an ideal. Commutative Algebra and Its Applications. Walter de Gruyter, Berlin, 2009, pp. 155–172.
A. J. Diesl: Nil clean rings. J. Algebra 383 (2013), 197–211.
J. Han, W.K. Nicholson: Extensions of clean rings. Commun. Algebra 29 (2001), 2589–2595.
Y. Hirano, H. Tominaga, A. Yaqub: On rings in which every element is uniquely expressible as a sum of a nilpotent element and a certain potent element. Math. J. Okayama Univ. 30 (1988), 33–40.
J. A. Huckaba: Commutative Rings with Zero-Divisors. Monographs and Textbooks in Pure and Applied Mathematics 117. Marcel Dekker, New York, 1988.
S.-E. Kabbaj: Matlis’ semi-regularity and semi-coherence in trivial ring extensions: A survey. To appear in Moroccan J. Algebra Geometry Appl.
M. Kabbour. Trivial ring extensions and amalgamations of periodic rings. Gulf J. Math. 3 (2015), 12–16.
N. Mahdou, M. A. S. Moutui: On (A)-rings and strongly (A)-rings issued from amalgamation. Stud. Sci. Math. Hung. 55 (2018), 270–279.
N. Mahdou, M. A. S. Moutui: Gaussian and Prüfer conditions in bi-amalgamated algebras. Czech. Math. J. 70 (2020), 381–391.
N. Mahdou, M. A. S. Moutui: Prüfer property in amalgamated algebras along an ideal. Ric. Mat. 69 (2020), 111–120.
W. W. McGovern, R. Raphael. Considering semi-clean rings of continuous functions. Topology Appl. 190 (2015), 99–108.
W. K. Nicholson. Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229 (1977), 269–278.
M. Ôhori. On strongly π-regular rings and periodic rings. Math. J. Okayama Univ. 27 (1985), 49–52.
J. Šter. Corner rings of a clean ring need not be clean. Commun. Algebra 40 (2012), 1595–1604.
Y. Ye. Semiclean rings. Commun. Algebra 31 (2003), 5609–5625.
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The authors would like to express their sincere thanks to the referee for his/her helpful suggestions and comments.
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Bakkari, C., Es-Saidi, M., Mahdou, N. et al. Extension of semiclean rings. Czech Math J 72, 461–476 (2022). https://doi.org/10.21136/CMJ.2021.0538-20
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DOI: https://doi.org/10.21136/CMJ.2021.0538-20