One-sided duo property on nilpotents

Year 2020, Volume: 49 Issue: 6, 1974 - 1987, 08.12.2020

Chan Yong Hong Hong Kee Kim Nam Kyun Kim , Tai Keun Kwak , Yang Lee

https://doi.org/10.15672/hujms.571016

Cited By: 1

Abstract

We study the structure of nilpotents in relation with a ring property that is near to one-sided  duo rings. Such a property is said to be one-sided nilpotent-duo. We prove the following for a one-sided nilpotent-duo  ring $R$: (i) The set of nilpotents in $R$ forms a subring; (ii) Köthe's conjecture holds for $R$; (iii) the subring generated by the identity and the set of nilpotents in $R$ is a  one-sided  duo ring; (iv) if the polynomial ring $R[x]$ over $R$ is  one-sided  nilpotent-duo then the set of nilpotents in $R$ forms a commutative ring, and $R[x]$ is an NI ring.  Several connections between  one-sided  nilpotent-duo and  one-sided duo are given. The structure of one-sided nilpotent-duo rings is also studied in various situations in ring theory. Especially we investigate several kinds of conditions under which  one-sided  nilpotent-duo rings are NI.

Keywords

right (left) nilpotent-duo ring, nilpotent, right (left) duo ring, radical, NI ring, matrix ring, ascending chain condition on left (right) annihilators

References

• [1] H.E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2, 363–368, 1970.
• [2] H.E. Bell and Y. Li, Duo group rings, J. Pure Appl. Algebra, 209, 833–838, 2007.
• [3] H.H. Brungs, Three questions on duo rings, Pacific J. Math. 58, 345–349, 1975.
• [4] A.W. Chatters and C.R. Hajarnavis, Rings with Chain Conditions, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1980.
• [5] Y.W. Chung and Y. Lee, Structures concerning group of units, J. Korean Math. Soc. 54, 177–191, 2017.
• [6] R.C. Courter, Finite dimensional right duo algebras are duo, Proc. Amer. Math. Soc. 84, 157–161, 1982.
• [7] J.L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38, 85–88, 1932.
• [8] E.H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89, 79–91, 1958.
• [9] K.R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979.
• [10] K.R. Goodearl and R.B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts 16, Cambridge University Press, Cambridge, 1989.
• [11] I.N. Herstein and L.W. Small, Nil rings satisfying certain chain conditions, Canad. J. Math. 16, 771–776, 1964.
• [12] I.N. Herstein and L.W. Small, Addendum to “Nil rings satisfying certain chain conditions", Canad. J. Math. 18, 300–302, 1966.
• [13] C. Huh, H.K. Kim and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra, 16, 37–52, 2002.
• [14] C. Huh, Y. Lee and A. Smoktunowicz, Armendariz rings and semicommutative ring, Comm. Algebra, 30, 751–761, 2002.
• [15] S.U. Hwang, Y.C. Jeon and Y. Lee, Structure and topological conditions of NI rings, J. Algebra , 302, 186–199, 2006.
• [16] H.K. Kim, N.K. Kim and Y. Lee, Weakly duo rings with nil Jacobson radical, J. Korean Math. Soc. 42, 455-468, 2005.
• [17] C. Lanski, Nil subrings of Goldie rings are nilpotent, Canad. J. Math. 21, 904–907, 1969.
• [18] B. Li, On potent rings, Commun. Korean Math. Soc. 23, 161–167, 2008.
• [19] G. Marks, On 2-primal Ore extensions, Comm. Algebra, 29, 2113–2123, 2001.
• [20] G. Marks, Duo rings and Ore extensions, J. Algebra, 280, 463–471, 2004.
• [21] J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons Ltd., Chichester, New York, Brisbane, Toronto, Singapore, 1987.
• [22] W.K. Nicholson, I-rings, Trans. Amer. Math. Soc. 207, 361–373, 1975.
• [23] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 , 269–278, 1977.
• [24] C. Polcino Milies and S.K. Sehgal, An Introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, 2002.
• [25] G. Thierrin, On duo rings, Canad. Math. Bull. 3, 167–172, 1960.
• [26] X. Yao, Weakly right duo rings, Pure Appl. Math. Sci. 21, 19–24, 1985.