A generalization of π-regular rings

We introduce the class of so-called regularly nil clean rings and systematically study their fundamentalcharacteristic properties accomplished with relationships among certain classical sorts of rings such as exchange rings,Utumi rings etc. These rings of ours naturally generalize the long-known classes of π-regular and strongly π-regularrings. We show that the regular nil cleanness possesses a symmetrization which extends the corresponding one forstrong π -regularity that was visualized by Dischinger [10]. Likewise, our achieved results substantially improve onestablishments presented in two recent papers by Danchev and Šter [8] and Danchev [6].

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[1] Arens R, Kaplansky I. Topological representation of algebras. Trans Amer Math Soc 1948; 63: 457-481.
[2] Azumaya G. Strongly -regular rings. J Fac Sci Hokkaido Univ Ser I 1954; 13: 34-39.
[3] Badawi A, Chin AYM, Chen HV. On rings with near idempotent elements. Internat J Pure & Appl Math 2002; 1: 253-259.
[4] Chin AYM. A note on strongly -regular rings. Acta Math Hungar 2004; 102: 337-342.
[5] Danchev PV. Invo-regular unital rings. Ann Univ Mariae Curie-Sklodowska Sect A - Math 2018; 72: 45-53.
[6] Danchev PV. Generalizing nil clean rings. Bull Belg Math Soc (Simon Stevin) 2018; 25: 13-28.
[7] Danchev PV, Lam TY. Rings with unipotent units. Publ Math Debrecen 2016; 88: 449-466.
[8] Danchev PV, Šter J. Generalizing -regular rings. Taiwanese J Math 2015; 19: 1577-1592.
[9] Diesl AJ. Nil clean rings. J Algebra 2013; 383: 197-211.
[10] Dischinger F. Sur les anneaux fortement -réguliers. C R Acad Sci Paris (Math) 1976; 283: 571-573 (in French).
[11] Drazin MP. Rings with central idempotent or nilpotent elements. Proc Edinburgh Math Soc 1958; 9: 157-165.
[12] Goodearl KR. von Neumann Regular Rings. Monographs and Studies in Mathematics, vol 4. Pitman (Advanced Publishing Program), Boston, Massachussetts, 1979.
[13] Kaplansky I. Topological representation of algebras II. Trans Amer Math Soc 1950; 68: 62-75.
[14] Khurana D, Lam TY, Nielsen PP. Two-sided properties of elements in exchange rings. Algebras & Represent Theory 2015; 18: 931-940.
[15] Khurana D, Lam TY, Wang Z. Rings of square stable range one. J Algebra 2011; 338: 122-143.
[16] Kosan T, Wang Z, Zhou Y. Nil-clean and strongly nil-clean rings. J Pure Appl Algebra 2016; 220: 633-646.
[17] Lam TY. A First Course in Noncommutative Rings, 2nd ed. Graduate Texts in Math Vol 131, Berlin, Germany: Springer-Verlag, 2001.
[18] McCoy NH. Generalized regular rings. Bull Amer Math Soc 1939; 45: 175-178.
[19] von Neumann J. On regular rings. Proc Nat Acad Sci (USA) 1936; 22: 101-108.
[20] Nicholson WK. Lifting idempotents and exchange rings. Trans Amer Math Soc 1977; 229: 269-278.
[21] Nicholson WK. Strongly clean rings and Fitting’s lemma. Commun Algebra 1999; 27: 3583-3592.
[22] Rowen LH. Finitely presented modules over semiperfect rings. Proc Amer Math Soc 1986; 97: 1-7.
[23] Rowen LH. Examples of semiperfect rings. Israel J Math 1989; 65: 273-283.
[24] Šter J. Weakly clean rings. J Algebra 2014; 401: 1-12.
[25] Šter J. On expressing matrices over Z2 as the sum of an idempotent and a nilpotent. Lin Algebra Appl 2018; 544: 339-349.
[26] Tuganbaev A. Rings Close to Regular, Mathematics and Its Applications, vol. 545. Dordrecht, the Netherlands: Kluwer Academic Publishers, 2002.
[27] Utumi J. On -rings. Proc Jap Acad Ser A (Math) 1957; 33: 63-66.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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