Jacobson investigated the structure of rings with the property that some power of each element is central, in the procedure of the study of commutativity. In this paper, we consider a class of rings in which this property occurs only for central elements, and such rings are called ppnc. We first prove that a noncommutative ppnc ring is infinite, and that a ppnc ring is commutative when it is a K-ring or a locally finite ring. We next study the structure of ppnc rings, and the relation between ppnc rings and related concepts (for example, K-rings, commutative rings and NI rings), through matrix rings, polynomial rings, right quotient rings and direct products.

Communicated by A. Leroy

Keywords:

AMSC: 16U70, 16U80

References

1. R. Antoine , Nilpotent elements and Armendariz rings, J. Algebra 319 (2008) 3128–3140. Crossref, Web of Science, Google Scholar
2. A. Bhattacharjee and U. S. Chakraborty , Ring endomorphisms satisfying the central reversible property, Proc. Math. Sci. 130(12) (2020) 12. Crossref, Web of Science, Google Scholar
3. K. J. Choi, T. K. Kwak and Y. Lee , Reversibility and symmetry over centers, J. Korean Math. Soc. 56 (2019) 723–738. Web of Science, Google Scholar
4. K. R. Goodearl , Von Neumann Regular Rings (Pitman, London, 1979). Google Scholar
5. C. Y. Hong and T. K. Kwak , On minimal strongly prime ideals, Comm. Algebra 28 (2000) 4867–4878. Crossref, Web of Science, Google Scholar
6. J. Huang, H.-L. Jin, T. K. Kwak, Y. Lee and Z. Piao , Remarks on centers of rings, Algebra Colloq. 28 (2021) 1–12. Link, Web of Science, Google Scholar
7. C. Huh, H. K. Kim and Y. Lee , P.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002) 37–52. Crossref, Web of Science, Google Scholar
8. C. Huh, N. K. Kim and Y. Lee , Examples of strongly $π$ -regular rings, J. Pure Appl. Algebra 189 (2004) 195–210. Crossref, Web of Science, Google Scholar
9. C. Huh, Y. Lee and A. Smoktunowicz , Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002) 751–761. Crossref, Web of Science, Google Scholar
10. T. W. Hungerford , Algebra (Springer-Verlag, New York, 1974). Google Scholar
11. S. U. Hwang, Y. C. Jeon and Y. Lee , Structure and topological conditions of NI rings, J. Algebra 302 (2006) 186–199. Crossref, Web of Science, Google Scholar
12. N. Jacobson , Structure of Rings, American Mathematical Society Colloquium Publications, Vol. 37 (American Mathematical Society, Providence, RI, 1964) [Revised edition]. Google Scholar
13. H.-L. Jin, T. K. Kwak, Y. Lee and Z. Piao , A property satisfying reducedness over centers, Algebra Colloq. 28 (2021) 453–468. Link, Web of Science, Google Scholar
14. D. A. Jordan , Bijective extensions of injective ring endomorphisms, J. Lond. Math. Soc. 25 (1982) 435–448. Crossref, Web of Science, Google Scholar
15. N. K. Kim, Y. Lee and S. J. Ryu , An ascending chain condition on Wedderburn radicals, Comm. Algebra 34 (2006) 37–50. Crossref, Web of Science, Google Scholar
16. J. Lambek , Lectures on Rings and Modules (Blaisdell Publishing Company, Waltham, 1966). Google Scholar
17. C. I. Lee and Y. Lee , Properties of K-rings and rings satisfying similar conditions, Int. J. Algebra Comput. 21 (2011) 1381–1394. Link, Web of Science, Google Scholar
18. Y. Lee , Structure of unit-IFP rings, J. Korean Math. Soc. 55 (2018) 1257–1268. Web of Science, Google Scholar
19. G. Marks , On 2-primal ore extensions, Comm. Algebra 29 (2001) 2113–2123. Crossref, Web of Science, Google Scholar
20. J. C. McConnell and J. C. Robson , Noncommutative Noetherian Rings (John Wiley & Sons Ltd., Chichester, New York, Brisbane, Toronto, Singapore, 1987). Google Scholar
21. L. H. Rowen , Ring Theory (Academic Press, San Diego, 1991). Google Scholar
22. G. Shin , Prime ideals and sheaf representation of a pseudosymmetric rings, Trans. Amer. Math. Soc. 184 (1973) 43–60. Crossref, Web of Science, Google Scholar