Abstract
Let \(f: A \longrightarrow B\) be a ring homomorphism and let J be an ideal of B. In this paper, we give a characterization for the amalgamation of A with B along J with respect to f (denoted \(A \bowtie ^{f}J\)) to be nil-clean.
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References
Anderson, D.D.: Commutative rings. In: Brewer, J., Glaz, S., Heinzer, W., Olberding, B. (eds.) Multiplicative Ideal Theory in Commutative Algebra: A Tribute to the Work of Robert Gilmer, pp. 1–20. Springer, New York (2006)
Boisen, M.B., Sheldon, P.B.: CPI-extension: over rings of integral domains with special prime spectrum. Can. J. Math. 29, 722–737 (1977)
Chhiti, M., Mahdou, N., Tamekkante, M.: Clean property in amalgamated algebras along an ideal. Hacettepe J. Math. Stat. 44(1), 41–49 (2015)
Danchev, P.V., McGovern, W.W.: Commutative weakly nil clean unital rings. J. Algebra 425, 410–422 (2015)
Diesl, A.J.: Nil clean rings. J. Algebra 383, 197–211 (2013)
Dorroh, J.L.: Concerning adjunctions to algebras. Bull. Am. Math. Soc. 38, 85–88 (1932)
D’Anna, M., Finocchiaro, C.A., Fontana, M.: Amalgamated Algebras Along an Ideal, Comm Algebra and Applications, pp. 241–252. Walter De Gruyter (2009)
D’Anna, M., Finocchiaro, C.A., Fontana, M.: Properties of chains of prime ideals in amalgamated algebras along an ideal. J. Pure Appl. Algebra 214, 1633–1641 (2010)
D’Anna, M.: A construction of Gorenstein rings. J. Algebra 306(2), 507–519 (2006)
D’Anna, M., Fontana, M.: The amalgamated duplication of a ring along a multiplicative canonical ideal. Ark. Mat. 45(2), 241–252 (2007)
D’Anna, M., Fontana, M.: An amalgamated duplication of a ring along an ideal: the basic properties. J. Algebra Appl. 6(3), 443–459 (2007)
Hirano, Y., Tominaga, H., Yaqub, A.: 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, 33–40 (1988)
Nagata, M.: Local Rings. Interscience, New York (1962)
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The authors would like to express their sincere thanks for the referee for his/her helpful suggestions and comments.
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Bakkari, C., Es-Saidi, M. Nil-clean property in amalgamated algebras along an ideal. Ann Univ Ferrara 65, 15–20 (2019). https://doi.org/10.1007/s11565-018-0304-8
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DOI: https://doi.org/10.1007/s11565-018-0304-8