Amalgamated rings with clean-type properties

Abstract

Let $f:A\rightarrow B$ be a ring homomorphism and $K$ be an ideal of $B$. Many variations of the notions of clean and nil-clean rings have been studied by a variety of authors. We investigate strongly $\pi$-regular and clean-like properties of the amalgamation ring $A\bowtie^{f}K$ of $A$ with $B$ along $K$ with respect to $f$.

Keywords

• amalgamation ring
• nil-clean ring
• J-clean ring
• semiclean ring
• semiregular ring
• exchange ring

References

1. [1] D.D. Anderson, D. Bennis, B. Fahid and A. Shaiea, On n-trivial extension of rings, Rocky Mountain J. Math. 47, 2439–2511, 2017.
2. [2] R. Antoine, Examples of Armendariz rings, Comm. Algebra, 38 (11), 4130–4143, 2010.
3. [3] C. Bakkari and M. Es-Saidi, Nil-clean property in amalgamated algebras along an ideal, Ann. Univ. Ferrara, 65, 15–20, 2019.
4. [4] M.B. Boisen and P.B. Sheldon, CPI-extension: Over rings of integral domains with special prime spectrum, Canad. J. Math. 29, 722–737, 1977.
5. [5] G. Călugăreanu, UU rings, Carpathian J. Math. 31 (2), 157–163, 2015.
6. [6] H. Chen, On strongly J-clean rings, Comm. Algebra, 38 (10), 3790–3804, 2010.
7. [7] M. Chhiti, N. Mahdou and M. Tamekkante, Clean property in amalgamated algebras along an ideal, Hacet. J. Math. Stat. 44 (1), 41–49, 2015.
8. [8] Y. Chun, Y.C. Jeon, S. Kang, K.N. Lee and Y. Lee, A concept unifying the Armendariz and NI conditions, Bull. Korean Math. Soc. 48 (1), 115–127, 2011.

9. [9] P. Crawley and B. Jónsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14, 797–855, 1964.
10. [10] P. Danchev and T.Y. Lam, Rings with unipotent units, Publ. Math. Debrecen, 88 (3-4), 449–466, 2016.
11. [11] M. D’Anna, A construction of Gorenstein rings, J. Algebra, 306 (2), 507–519, 2006.
12. [12] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2), 241–252, 2007.
13. [13] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 443–459, 2007.
14. [14] M. D’Anna, C. A. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, Commutative algebra and its applications, Walter de Gruyter, Berlin, 241–252, 2009.
15. [15] A.J. Diesl, Nil clean rings, J. Algebra, 383, 197–211, 2013.
16. [16] M.F. Dischinger, Sur les anneaux fortement π-reguliers, C. R. Acad. Sc. Paris, 283, 571–573, 1976.
17. [17] C.Y. Honga, N. Kimb and Y. Lee, Exchange rings and their extensions, J. Pure Appl. Algebra, 179, 117–126, 2003.
18. [18] T.Y. Lam, A first course in noncommutative rings, Berlin-Heidelberg-New York: Springer-Verlag, 1991.
19. [19] M. Nagata, Local Rings, Interscience, New York, 1962.
20. [20] W.K. Nicholson, Semiregular modules and rings, Canad. J. Math. XXVIII, 1105–1120, 1976.
21. [21] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229, 269–278, 1977.
22. [22] W.K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra, 27 (8), 3583–3592, 1999.
23. [23] W.K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit. Glasgow Math. J. 46, 227–236, 2004.
24. [24] S. Sahinkaya, G. Tang and Y. Zhou, Nil-clean group rings, J. Algebra Appl. 16 (5), 1750135, 2017.
25. [25] J. Stock, On rings whose projective modules have the exchange property, J. Algebra, 103, 437–453, 1986.
26. [26] A. Tuganbaev, Rings close to regular, Moscow Power Engineering Institute, Technological University, Moscow, Russia 2002.
27. [27] R.B. Warfield Jr., Exchange rings and decompositions of modules, Math. Ann. 199, 31–36, 1972.
28. [28] Y. Ye, Semiclean rings, Comm. Algebra, 31 (11), 5609–5625, 2003.