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STRUCTURE OF IDEMPOTENTS IN RINGS WITHOUT IDENTITY
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 Title & Authors
STRUCTURE OF IDEMPOTENTS IN RINGS WITHOUT IDENTITY
Kim, Nam Kyun; Lee, Yang; Seo, Yeonsook;
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 Abstract
We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that -regular rings are strongly -regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.
 Keywords
idempotent;right IIP ring;right IR ring;Abelian ring;
 Language
English
 Cited by
1.
Ring properties related to symmetric rings, International Journal of Algebra and Computation, 2014, 24, 07, 935  crossref(new windwow)
 References
1.
D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852. crossref(new window)

2.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.

3.
G. Azumaya, Strongly ${\pi}$-regular rings, J. Fac. Sci. Hokkaido Univ. Ser. I. 13 (1954), 34-39.

4.
H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368. crossref(new window)

5.
H. E. Bell, A commutativity study for periodic rings, Pacific J. Math. 70 (1977), no. 1, 29-36. crossref(new window)

6.
P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648. crossref(new window)

7.
F. Dischinger, Sur les anneaux fortement ${\pi}$-reguliers, C. R. Acad. Sci. Paris Ser. A-B 283 (1976), no. 8, 571-573.

8.
K. E. Eldridge, Orders for finite noncommutative rings with unity, Amer. Math. Monthly 75 (1968), no. 5, 512-514. crossref(new window)

9.
D. B. Erickson, Orders for finite noncommutative rings, Amer. Math. Monthly 73 (1966), 376-377. crossref(new window)

10.
E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89 (1958), 79-91. crossref(new window)

11.
C. Huh, H. K. Kim, N. K. Kim, and Y. Lee, Basic examples and extensions of symmetric rings, J. Pure Appl. Algebra 202 (2005), no. 1-3, 154-167. crossref(new window)

12.
C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37-52. crossref(new window)

13.
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. crossref(new window)

14.
N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223. crossref(new window)

15.
R. L. Kruse and D. T. Price, Nilpotent Rings, Gordon and Breach, New York, London, Paris, 1969.

16.
J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.

17.
J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368. crossref(new window)

18.
G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra 174 (2002), no. 3, 311-318. crossref(new window)

19.
L. Motais de Narbonne, Anneaux semi-commutatifs et uniseriels anneaux dont les ideaux principaux sont idempotents, In: Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71-73, Bib. Nat., Paris 1982.

20.
G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60. crossref(new window)

21.
W. Xue, Structure of minimal noncommutative duo rings and minimal strongly bounded nonduo rings, Comm. Algebra 20 (1992), no. 9, 2777-2788. crossref(new window)

22.
L. Xu and W. Xue, Structure of minimal non-commutative zero-insertive rings, Math. J. Okayama Univ. 40 (1998), 69-76.