IDEMPOTENTS IN π-REGULAR RINGS, RIGHT AI RINGS, NI RINGS AND GENERALIZED REGULAR RINGS
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Von Neumann regular rings are studied by ring theorists and functional analysts in connection with operator algebra theory. In particular, the concept of idempotent in algebra is a generalization of projection in analysis. We study the structure of idempotents in $\pi$-regular rings, right AI rings (i.e., for every element $a$, $ab$ is an idempotent for some nonzero element $b$), NI rings, and generalized regular rings (i.e., every nonzero principal right ideal contains a nonzero idempotent). We obtain a well-known fact, proved by Menal, Nicholson and Zhou, that idempotents can be lifted modulo every ideal in $\pi$-regular rings, as a corollary of one of main results of this article. It is shown that the $\pi$-regularity is seated between right AI and regularity. We also show that from given any $\pi$-regular ring, we can construct a right AI ring but not $\pi$-regular. In addition, we study the structure of idempotents of $\pi$-regular rings and right AI rings in relation to the ring properties of Abelian and NI, giving simpler proofs to well-known results for Abelian $\pi$-regular rings.
Abstract
Von Neumann regular rings are studied by ring theorists and functional analysts in connection with operator algebra theory. In particular, the concept of idempotent in algebra is a generalization of projection in analysis. We study the structure of idempotents in $\pi$-regular rings, right AI rings (i.e., for every element $a$, $ab$ is an idempotent for some nonzero element $b$), NI rings, and generalized regular rings (i.e., every nonzero principal right ideal contains a nonzero idempotent). We obtain a well-known fact, proved by Menal, Nicholson and Zhou, that idempotents can be lifted modulo every ideal in $\pi$-regular rings, as a corollary of one of main results of this article. It is shown that the $\pi$-regularity is seated between right AI and regularity. We also show that from given any $\pi$-regular ring, we can construct a right AI ring but not $\pi$-regular. In addition, we study the structure of idempotents of $\pi$-regular rings and right AI rings in relation to the ring properties of Abelian and NI, giving simpler proofs to well-known results for Abelian $\pi$-regular rings.
참고문헌 (22)
- D. D. Anderson, "Armendariz rings and Gaussian rings", Comm. Algebra, 26(7), pp.2265~2272 (1998)[DOI]
- A. Badawi, "On abelian π-regular rings", Comm. Algebra, 25(4), pp.1009~1021 (1997)[DOI]
- G. F. Birkenmeier, "Completely prime ideals and associated radicals, Ring theory", pp.102~129 (1993), World Sci. Publ
- G. F. Birkenmeier, "A connection between weak regularity and the simplicity of prime factor rings", Proc. Amer. Math. Soc, 122(1), pp.53~58 (1994)[DOI]
- K. R. Goodearl, "von Neumann Regular Rings" (1979), Pitman
- J. Han, "Structure of Abelian rings", Front. Math. China, 12(1), pp.117~134 (2017)[DOI]
- J. Han, "Structure of NI rings related to centers", Comm. Algebra, 49(2), pp.621~630 (2021)[DOI]
- J. Han, "On idempotents in relation with regularity", J. Korean Math. Soc, 53(1), pp.217~232 (2016)[DOI]
- C. Y. Hong, "On minimal strongly prime ideals", Comm. Algebra, 28(10), pp.4867~4878 (2000)[DOI]
- C. Huh, "p. p. rings and generalized p. p. rings", J. Pure Appl. Algebra, 167(1), pp.37~52 (2002)[DOI]
- S. U. Hwang, "Structure and topological conditions of NI rings", J. Algebra, 302(1), pp.186~199 (2006)[DOI]
- J. Lambek, "Lectures on Rings and Modules" (1966), Blaisdell Publishing Co.
- Y. Lee, "A note on π-regular rings", Kyungpook Math. J, 38(1), pp.157~161 (1998)
- G. Marks, "On 2-primal Ore extensions", Comm. Algebra, 29(5), pp.2113~2123 (2001)[DOI]
- J. C. McConnell, "Noncommutative Noetherian Rings, Pure and Applied Mathematics" (1987), John Wiley & Sons, Ltd.
- N. H. McCoy, "Generalized regular rings", Bull. Amer. Math. Soc, 45(2), pp.175~178 (1939)[DOI]
- P. Menal, "On π-regular rings whose primitive factor rings are Artinian", J. Pure Appl. Algebra, 20(1), pp.71~78 (1981)[DOI]
- W. K. Nicholson, "Strong lifting", J. Algebra, 285(2), pp.795~818 (2005)[DOI]
- M. Ohori, "On strongly π-regular rings and periodic rings", Math. J. Okayama Univ, 27, pp.49~52 (1985)
- G. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Trans. Amer. Math. Soc, 184, pp.43~60 (1973)[DOI]
- J. von Neumann, "Zur algebra der funktionaloperationen und theorie der normalen operatoren", Math. Ann, 102(1), pp.370~427 (1930)[DOI]
- J. von Neumann, "On regular rings", Proceedings of the National Academy of Sciences, 22, pp.707~713 (1936)