Document Type : Original Manuscript

Authors

1 Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.

2 Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran.

Abstract

We say a ring R with unity is left weakly Baer if the left annihilator
of any nonempty subset of R is right s-unital by right semicentral idempotents,
which implies that R modulo the left annihilator of any nonempty subset is
flat. It is shown that, unlike the Baer or right PP conditions, the weakly
Baer property is inherited by polynomial extensions. Examples are provided
to explain the results.

Keywords

1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra, 26(7) (1998), 2265–2272.
2. E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc., 18 (1974), 470–473.
3. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Aust. Math. Soc., 2 (1970), 363–368.
4. G. F. Birkenmeier, Idempotents and completely semiprime ideals, Comm. Algebera, 11 (1983), 567–580.
5. G. F. Birkenmeier, J. Y. Kim and J. K. Park, A sheaf representation of quasi-Baer rings, J. Pure. Appl. Algebra, 146 (2000), 209–223.
6. G. F. Birkenmeier, J. Y. Kim and J. K. Park, On quasi-Baer rings, Contemp. Math., 259 (2000), 67–92.
7. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra, 29(2) (2001), 639–660.
8. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Quasi-Baer ring extensions and biregular rings, Bull. Aust. Math. Soc., 61 (2000), 39–52.
9. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Semicentral reduced algebras, in International Symp. Ring Theory, eds. G. F. Birkenmeier, J. K. Park and Y. S. Park, Birkhauser, Boston, 2001.
10. G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra, 265(2) (2003), 457–477.
11. G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Extensions of Rings and Modules, Birkh¨auser, New York, 2013.
12. S. A. Chase, Generalization of triangular matrices, Nagoya Math. J., 18 (1961), 13–25.
13. W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J., 34 (1967), 417–423.
14. S. Endo, Note on P.P. rings, Nagoya Math. J., 17 (1960), 167–170.
15. K. R. Goodearl, Von Neumann Regular Rings, Krieger, Malabar, 1991.
16. Y. Hirano, On annihilator ideals of a polynomail ring over noncommutative ring, J. Pure Appl. Algebra, 168(1) (2002), 45–52.
17. I. Kaplansky, Rings of Operators, Benjamin, New York, 1965.
18. Y. Lee, N. K. Kim and C. Y. Hong, Counterexamples on baer rings, Comm. Algebra, 25(2) (1997), 497–507.
19. Z. Liu and R. Zhao, A generalization of PP-rings and p.q.-Baer rings, Glasgow Math. J., 48(2) (2006), 217–229.
20. A. Majidinya, A. Moussavi, Weakly principally quasi-Baer rings, J. Algebra Appl., 15(1) (2016), Article ID: 1650002 .
21. A. Majidinya, A. Moussavi and K. Paykan, Generalized APP-rings, Comm. Algebra, 41(12) (2013), 4722–4750.
22. A. C. Mewborn, Regular rings and Baer rings, Math. Z., 121 (1971), 211–219.
23. A. R. Nasr-Isfahani, A. Moussavi, On ore extensions of quasi-Baer rings, J. Algebra Appl., 7(2) (2008), 211–224.
24. A. Pollingher and A. Zaks, On Baer and quasi-Baer rings, Duke Math. J., 37 (1970), 127–138.
25. C. E. Rickart, Banach algebras with an adjoint operation, Ann. of Math., 47(2) (1946), 528–550.
26. S. T. Rizvi and C. S. Roman, Baer and quasi Baer modules, Comm. Algebra, 32(1) (2004), 103–123.
27. L. W. Small, Semihereditary rings, Bull. Amer. Math. Soc., 73 (1967), 656–658.
28. B. Stenstrom, Rings of Quotients, Springer-Verlag, Berlin, Heidelberg, 1975.
29. H. Tominaga, On s-unital rings, Math. J. Okayama Univ., 18 (1976), 117–134.