Bulletin of the Korean Mathematical Society (대한수학회보)

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ADDITIVE MAPS OF SEMIPRIME RINGS SATISFYING AN ENGEL CONDITION

  • Lee, Tsiu-Kwen (Department of Mathematics National Taiwan University) ;
  • Li, Yu (School of Mathematics and Statistics Southwest University) ;
  • Tang, Gaohua (School of Sciences Beibu Gulf University)
  • Received : 2020.05.21
  • Accepted : 2020.11.02
  • Published : 2021.05.31
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Abstract

Let R be a semiprime ring with maximal right ring of quotients Qmr(R), and let n1, n2, …, nk be k fixed positive integers. Suppose that R is (n1+n2+⋯+nk)!-torsion free, and that f : 𝜌 → Qmr(R) is an additive map, where 𝜌 is a nonzero right ideal of R. It is proved that if [[…[f(x), xn1], …], xnk] = 0 for all x ∈ 𝜌, then [f(x), x] = 0 for all x ∈ 𝜌. This gives the result of Beidar et al. [2] for semiprime rings. Moreover, it is also proved that if R is p-torsion, where p is a prime integer with p = Σki=1 ni and if f : R → Qmr(R) is an additive map satisfying [[…[f(x), xn1], …], xnk] = 0 for all x ∈ R, then [f(x), x] = 0 for all x ∈ R.

Keywords

Acknowledgement

The work of G. H. Tang was supported by the national natural science foundation of China (11661014, 11961050, 11661013), the work of T.-K. Lee was supported in part by the Ministry of Science and Technology of Taiwan (MOST 107-2115-M-002-018-MY2).

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