Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AN ASSOCIATED SEQUENCE OF IDEALS OF AN INCREASING SEQUENCE OF RINGS

  • Ali, Benhissi (Mathematics of Department Faculty of Sciences of Monastir Monastir University) ;
  • Abdelamir, Dabbabi (Mathematics of Department Faculty of Sciences of Monastir Monastir University)
  • Received : 2021.06.01
  • Accepted : 2022.08.01
  • Published : 2022.11.30
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Abstract

Let 𝒜 = (An)n≥0 be an increasing sequence of rings. We say that 𝓘 = (In)n≥0 is an associated sequence of ideals of 𝒜 if I0 = A0 and for each n ≥ 1, In is an ideal of An contained in In+1. We define the polynomial ring and the power series ring as follows: $I[X]\, = \,\{\, f \,=\, {\sum}_{i=0}^{n}a_iX^i\,{\in}\,A[X]\,:\,n\,{\in}\,\mathbb{N},\,a_i\,{\in}\,I_i \,\}$ and $I[[X]]\, = \,\{\, f \,=\, {\sum}_{i=0}^{+{\infty}}a_iX^i\,{\in}\,A[[X]]\,:\,a_i\,{\in}\,I_i \,\}$. In this paper we study the Noetherian and the SFT properties of these rings and their consequences.

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References

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