Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T06:54:46.032Z Has data issue: false hasContentIssue false
  • English
  • Français

ON RINGS WHOSE RIGHT ANNIHILATORS ARE BOUNDED

Published online by Cambridge University Press:  01 September 2009

SEO UN HWANG ,
NAM KYUN KIM  and
YANG LEE
SEO UN HWANG
Affiliation:
Department of Mathematics, Busan National University, Busan 609-735, Korea e-mail: hwangseo@dreamwiz.com
NAM KYUN KIM
Affiliation:
College of Liberal Arts, Hanbat National University, Daejeon 305-719, Korea e-mail: nkkim@hanbat.ac.kr
YANG LEE
Affiliation:
Department of Mathematics Education, Busan National University, Pusan 609-735, Korea e-mail: ylee@pusan.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterexamples to several naturally raised situations in the process.

Keywords

16D2516D4016E50
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009 , pp. 539 - 559
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Anderson, D. D. and Camillo, V., Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), 2265–2272.CrossRefGoogle Scholar
2.Anderson, D. D. and Camilo, V., Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), 2847–2852.CrossRefGoogle Scholar
3.Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer-Verlag, New York, 1992).CrossRefGoogle Scholar
4.Armendariz, E. P., A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470–473.CrossRefGoogle Scholar
5.Armendariz, E. P., Rings with DCC on essential left ideals, Comm. Algebra 8 (1980), 299–308.CrossRefGoogle Scholar
6.Bell, H. E., Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363–368.CrossRefGoogle Scholar
7.Birkenmeier, G. F. and Tucci, R. P., Homomorphic images and the singular ideal of a strongly right bounded ring, Comm. Algebra 16 (1988), 1099–1122.CrossRefGoogle Scholar
8.Chatters, A. W. and Xue, W., On right duo p.p. rings, Glasgow Math. J. 32 (1990), 221–225.CrossRefGoogle Scholar
9.Chen, J., On von Neumann regular rings and SF-rings, Math. Japon. 36 (1991), 1123–1127.Google Scholar
10.Courter, R. C., Finite dimensional right duo algebras are duo, Proc. Am. Math. Soc. 84 (1982), 157–161.Google Scholar
11.de Narbonne, L. M., Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents. In Proceedings of the 106th National Congress of Learned Societies, (Bibliotheque Nationale, Paris, 1982), 71–73.Google Scholar
12.Eldridge, K. E., Orders for finite noncommutative rings with unity, Am. Math. Monthly 73 (1966), 376–377.Google Scholar
13.Faith, C., Algebra II ring theory (Springer-Verlag, Berlin, 1976).CrossRefGoogle Scholar
14.Feller, E. H., Properties of primary noncommutative rings, Trans. Am. Math. Soc., 89 (1958), 79–91.CrossRefGoogle Scholar
15.Goodearl, K. R., Von Neumann regular rings (Pitman, London, 1979).Google Scholar
16.Hirano, Y., On rings whose simple modules are flat, Can. Math. Bull. 37 (1994), 361–364.CrossRefGoogle Scholar
17.Hirano, Y., Hong, C. Y., Kim, J. Y. and Park, J. K., On strongly bounded rings and duo rings, Comm. Algebra 23 (1995), 2199–2214.CrossRefGoogle Scholar
18.Hong, C. Y. and Kwak, T. K., On minimal strongly prime ideals, Comm. Algebra 28 (2000), 4867–4878.CrossRefGoogle Scholar
19.Huh, C., Kim, H. K. and Lee, Y., P.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), 37–52.CrossRefGoogle Scholar
20.Huh, C., Lee, Y. and Smoktunowicz, A., Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), 751–761.CrossRefGoogle Scholar
21.Jacobson, N., The theory of rings (American Mathematical Society, Providence, RI, 1943).CrossRefGoogle Scholar
22.Jacobson, N., Structure of rings, 2nd ed., Colloquium Publication 37 (American Mathematical Society, 1964).Google Scholar
23.Kim, N. K. and Lee, Y., Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), 207–223.CrossRefGoogle Scholar
24.Köthe, G., Die Struktur der Ringe deren Restklassenring nach dem Radikal vollständig reduzibel ist, Math. Z., 42 (1930), 161–186.CrossRefGoogle Scholar
25.Kruse, R. and Price, D., Nilpotent rings (Gordon and Breach, New York, 1969).Google Scholar
26.Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, MA/Toronto/London, 1966).Google Scholar
27.Marks, G., On 2-primal Ore extensions, Comm. Algebra 29 (2001), 2113–2123.CrossRefGoogle Scholar
28.Nakayama, T., On Frobeniusean algebras. I. Ann. Math. (2) 40 (1939), 611–633.CrossRefGoogle Scholar
29.Ramamurthi, V. S., On the injectivity and flatness of certain cyclic modules, Proc. Am. Math. Soc. 48 (1975), 21–25.CrossRefGoogle Scholar
30.Rege, M. B., On von Neumann regular rings and SF-rings, Math. Japon. 31 (1986), 927–936.Google Scholar
31.Rege, M. B. and Chhawchharia, S., Armendariz rings, Proc. Jpn. Acad. Ser. A Math. Sci. 73 (1997), 14–17.CrossRefGoogle Scholar
32.Shin, G., Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Am. Math. Soc. 84 (1973), 43–60.CrossRefGoogle Scholar
33.Xiao, Y., SF-rings and exellent extensions, Comm. Algebra 22 (1994), 2463–2471.CrossRefGoogle Scholar
34.Xue, W., On strongly right bounded finite rings, Bull. Austral. Math. Soc. 44 (1991), 353–355.CrossRefGoogle Scholar
35.Xue, W., Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings, Comm. Algebra 20 (1992), 2777–2788.CrossRefGoogle Scholar
36.Yu, H.-P., On quasi-duo rings, Glasgow Math. J. 37 (1995), 21–31.CrossRefGoogle Scholar
37.Yue Chi Ming, R., Maximal ideals in regular rings, Hokkaido Math. J. 12 (1988), 119–128.Google Scholar
38.Zhang, J. and Du, X., Von Neumann regularity of SF-rings, Comm. Algebra 21 (1993), 2445–2451.Google Scholar