Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Ore Extension Rings with Constant Products of Elements

  • Hashemi, Ebrahim (Faculty of Mathematical Sciences, Shahrood University of Technology) ;
  • Alhevaz, Abdollah (Faculty of Mathematical Sciences, Shahrood University of Technology)
  • Received : 2018.04.07
  • Accepted : 2018.11.20
  • Published : 2019.12.23
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Abstract

Let R be an associative unital ring with an endomorphism α and α-derivation δ. The constant products of elements in Ore extension rings, when the coefficient ring is reversible, is investigated. We show that if f(x) = ∑ni=0 aixi and g(x) = ∑mj=0 bjxj be nonzero elements in Ore extension ring R[x; α, δ] such that g(x)f(x) = c ∈ R, then there exist non-zero elements r, a ∈ R such that rf(x) = ac, when R is an (α, δ)-compatible ring which is reversible. Among applications, we give an exact characterization of the unit elements in R[x; α, δ], when the coeficient ring R is (α, δ)-compatible. Furthermore, it is shown that if R is a weakly 2-primal ring which is (α, δ)-compatible, then J(R[x; α, δ]) = N iℓ(R)[x; α, δ]. Some other applications and examples of rings with this property are given, with an emphasis on certain classes of NI rings. As a consequence we obtain generalizations of the many results in the literature. As the final part of the paper we construct examples of rings that explain the limitations of the results obtained and support our main results.

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References

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