RESEARCH ARTICLE

Generalizations of von Neumann regular rings, PP rings, and Baer rings

  • Lixin MAO
Expand
  • Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China

Received date: 30 Mar 2014

Accepted date: 29 Jan 2016

Published date: 18 Apr 2016

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We introduce the notions of IDS modules, IP modules, and Baer* modules, which are new generalizations of von Neumann regular rings, PPrings, and Baer rings, respectively, in a general module theoretic setting. We obtain some characterizations and properties of IDS modules, IP modules and Baermodules. Some important classes of rings are characterized in terms of IDS modules, IP modules, and Baer*modules.

Cite this article

Lixin MAO . Generalizations of von Neumann regular rings, PP rings, and Baer rings[J]. Frontiers of Mathematics in China, 2016 , 11(2) : 377 -400 . DOI: 10.1007/s11464-016-0523-1

1
Anderson F W, Fuller K R. Rings and Categories of Modules. New York: Springer-Verlag, 1974

DOI

2
Camillo V. Coherence for polynomial rings. J Algebra, 1990, 132: 72–76

DOI

3
Ding N Q, Chen J L. Relative coherence and preenvelopes. Manuscripta Math, 1993, 81: 243–262

DOI

4
Endo S. Note on p.p. rings. Nagoya Math J, 1960, 17: 167–170

5
Fieldhouse D J. Pure theories. Math Ann, 1969, 184: 1–18

DOI

6
Goodearl K R. Ring Theory: Nonsingular Rings and Modules. Monographs and Textbooks in Pure and Applied Mathematics, Vol 33. New York and Basel: Marcel Dekker, Inc, 1976

7
Hattori A. A foundation of torsion theory for modules over general rings. Nagoya Math J, 1960, 17: 147–158

8
Jødrup J. p.p. rings and finitely generated flat ideals. Proc Amer Math Soc, 1971, 28: 431–435

9
Jones M F. f-Projectivity and flat epimorphisms. Comm Algebra, 1981, 9: 1603–1616

DOI

10
Kaplansky I. Rings of Operators. New York: Benjamin, 1968

11
Lam T Y. Lectures on Modules and Rings. New York-Heidelberg-Berlin: Springer-Verlag, 1999

DOI

12
Lee G, Rizvi S T, Roman C S. Rickart modules. Comm Algebra, 2010, 38: 4005–4027

DOI

13
Lee G, Rizvi S T, Roman C S. Dual Rickart modules. Comm Algebra, 2011, 39: 4036–4058

DOI

14
Lee T K, Zhou Y Q. Reduced modules, rings, modules, algebras and abelian groups. In: Facchini A, Houston E, Salce L. eds. Rings, Modules, Algebras, and Abelian Groups. Lecture Notes in Pure and Appl Math, Vol 236. New York: Dekker, 2004, 365–377

15
Liu Q, Ouyang B Y. Rickart modules. Nanjing Daxue Xuebao Shuxue Bannian Kan, 2006, 23(1): 157–166

16
Liu Q, Ouyang B Y, Wu T S. Principally quasi-Baer modules. J Math Res Exposition, 2009, 29(5): 823–830

17
Mao L X. Rings close to Baer. Indian J Pure Appl Math, 2007, 38(3): 129–142

18
Mao L X. Properties of P-coherent and Baer modules. Period Math Hungar, 2010, 60(2): 97–114

DOI

19
Mao L X, Ding N Q. Relative flatness, Mittag-Leffler modules and endocoherence. Comm Algebra, 2006, 34(9): 3281–3299

DOI

20
Nicholson W K. On PP-rings. Period Math Hungar, 1993, 27(2): 85–88

DOI

21
Nicholson W K, Watters J F. Rings with projective socle. Proc Amer Math Soc, 1988, 102: 443–450

DOI

22
Rizvi S T, Roman C S. Baer property of modules and applications. In: Chen J L, Ding N Q, Marubayashi, H. eds. Advances in Ring Theory. 2005, Singapore: World Scientific, 225–241

DOI

23
Rotman J J. An Introduction to Homological Algebra. New York: Academic Press, 1979

24
Ware R. Endomorphism rings of projective modules. Trans Amer Math Soc, 1971, 155: 233–256

DOI

25
Wisbauer R. Foundations of Module and Ring Theory. Philadelphia: Gordon and Breach, 1991

26
Xue W M. On PP rings. Kobe J Math, 1990, 7: 77–80

27
Zelmanowitz J. Regular modules. Trans Amer Math Soc, 1972, 163: 340–355

DOI

28
Zhu H Y, Ding N Q. Generalized morphic rings and their applications. Comm Algebra, 2007, 35: 2820–2837

DOI

Outlines

/