$\pi$-BAER $\ast$-RINGS

Abstract

A $\ast$-ring $R$ is called a $\pi$-Baer $\ast$-ring, if for any projection invariant left ideal $Y$ of $R$, the right annihilator of $Y $ is generated, as a right ideal, by a projection. In this note, we study some properties of such $\ast$-rings. We indicate interrelationships between the $\pi$-Baer $\ast$-rings and related classes of rings such as $\pi$-Baer rings, Baer $\ast$-rings, and quasi-Baer $\ast$-rings. We announce several results on $\pi$-Baer $\ast$-rings. We show that this notion is well-behaved with respect to polynomial extensions and full matrix rings. Examples are provided to explain and delimit our results.

Keywords

• $\pi$-Baer ring
• $\pi$-Baer $\ast$-ring
• quasi-Baer $\ast$-ring
• Baer ring
• Baer $\ast$-ring

References

1. M. Ahmadi, N. Golestani and A. Moussavi, Generalized quasi-Baer $ \ast $-rings and Banach $ \ast $-algebras, Comm. Algebra, 48(5) (2020), 2207-2247.
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. Austral. Math. Soc., 2 (1973), 363-368.
4. S. K. Berberian, Baer $ \ast $-Rings, Grundlehren Math. Wiss., Vol. 195, Springer-Verlag, New York-Berlin, 1972.
5. G. F. Birkenmeier N. J. Groenewald and H. E. Heatherly, Minimal and maximal ideals in rings with involution, Beitrage Algebra Geom., 38(2) (1997), 217-225.
6. G. F. Birkenmeier, Y. Kara and A. Tercan, $\pi$-Baer rings, J. Algebra Appl., 17(2) (2018), 1850029 (19 pp).
7. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Quasi-Baer ring extensions and biregular rings, Bull. Austral. Math. Soc., 61(1) (2000), 39-52.
8. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra, 159(1) (2001), 25-42.

9. G. F. Birkenmeier, B. J. Muller and S. T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Comm. Algebra, 30(3) (2002), 1395-1415.
10. G. F. Birkenmeier and J. K. Park, Self-adjoint ideals in Baer $ \ast $-rings, Comm. Algebra, 28(9) (2000), 4259-4268.
11. G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra, 265(2) (2003), 457-477.
12. G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Hulls of semiprime rings withapplications to C*-algebras, J. Algebra, 322(2) (2009), 327-352.
13. G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Extensions of Rings and Modules, Birkhauser/Springer, New York, 2013.
14. K. A. Brown, The singular ideals of group rings, Quart. J. Math. Oxford Ser. (2), 28(109) (1977), 41-60.
15. W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J., 34 (1967), 417-423.
16. D. E. Handelman, Prufer domains and Baer $ \ast $-rings, Arch. Math. (Basel), 29(3) (1977), 241-251.
17. I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
18. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999.
19. W. Narkiewicz, Polynomial Mappings, Lecture Notes in Mathematics, 1600, Springer-Verlag, Berlin, 1995.