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A QUESTION ABOUT MAXIMAL NON φ-CHAINED SUBRINGS

  • Atul Gaur (Department of Mathematics University of Delhi) ;
  • Rahul Kumar (Department of Mathematics Birla Institute of Technology and Science Pilani)
  • Received : 2021.08.09
  • Accepted : 2022.06.15
  • Published : 2023.01.31
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Abstract

Let 𝓗0 be the set of rings R such that Nil(R) = Z(R) is a divided prime ideal of R. The concept of maximal non φ-chained subrings is a generalization of maximal non valuation subrings from domains to rings in 𝓗0. This generalization was introduced in [20] where the authors proved that if R ∈ 𝓗0 is an integrally closed ring with finite Krull dimension, then R is a maximal non φ-chained subring of T(R) if and only if R is not local and |[R, T(R)]| = dim(R) + 3. This motivates us to investigate the other natural numbers n for which R is a maximal non φ-chained subring of some overring S. The existence of such an overring S of R is shown for 3 ≤ n ≤ 6, and no such overring exists for n = 7.

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References

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