Frontiers of Mathematics in China >
Generalizations of von Neumann regular rings, PP rings, and Baer rings
Received date: 30 Mar 2014
Accepted date: 29 Jan 2016
Published date: 18 Apr 2016
Copyright
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 Baer∗ modules. Some important classes of rings are characterized in terms of IDS modules, IP modules, and Baer*modules.
Key words： IDS Module; IP module; Baer∗ module; von Neumann regular ring; PP ring; Baer ring
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
|
| 2 |
Camillo V. Coherence for polynomial rings. J Algebra, 1990, 132: 72–76
|
| 3 |
Ding N Q, Chen J L. Relative coherence and preenvelopes. Manuscripta Math, 1993, 81: 243–262
|
| 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
|
| 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
|
| 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
|
| 12 |
Lee G, Rizvi S T, Roman C S. Rickart modules. Comm Algebra, 2010, 38: 4005–4027
|
| 13 |
Lee G, Rizvi S T, Roman C S. Dual Rickart modules. Comm Algebra, 2011, 39: 4036–4058
|
| 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
|
| 19 |
Mao L X, Ding N Q. Relative flatness, Mittag-Leffler modules and endocoherence. Comm Algebra, 2006, 34(9): 3281–3299
|
| 20 |
Nicholson W K. On PP-rings. Period Math Hungar, 1993, 27(2): 85–88
|
| 21 |
Nicholson W K, Watters J F. Rings with projective socle. Proc Amer Math Soc, 1988, 102: 443–450
|
| 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
|
| 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
|
| 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
|
| 28 |
Zhu H Y, Ding N Q. Generalized morphic rings and their applications. Comm Algebra, 2007, 35: 2820–2837
|
/
| 〈 |
|
〉 |