The primeness of noncommutative polynomials on prime rings

KOŞAN M. T., Lee T.

Journal of Algebra and its Applications, vol.24, no.12, 2025 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 24 Issue: 12
  • Publication Date: 2025
  • Doi Number: 10.1142/s0219498825502767
  • Journal Name: Journal of Algebra and its Applications
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Keywords: extended centroid, GPI, higher commutator, idempotent, noncommutative polynomial, PI, Prime ring, X-prime
  • Gazi University Affiliated: Yes

Abstract

We study the primeness of noncommutative polynomials on prime rings. Let R be a prime ring with extended centroid C, ρ a right ideal of R, f(X1,..,Xt) a noncommutative polynomial over C, which is not a polynomial identity (PI) for ρ, and a,b €-{0}. Then af(x1,..,xt)b = 0 for all x1,..,xt ρ if and only if one of the following holds: (i) aρ = 0; (ii) ρC = eRC for some idempotent e RC and b ρC such that either f(X1,..,Xt)Xt+1 is a PI for ρ or f(X1,..,Xt) is central-valued on eRCe and ab = 0. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of X-primeness of rings.