• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.663-677
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• 2023

The purpose of this paper is to introduce a new class of rings containing the class of SFT-rings and contained in the class of rings with Noetherian prime spectrum. Let A be a commutative ring with unit and I be an ideal of A. We say that I is SFT if there exist an integer k ≥ 1 and a finitely generated ideal F ⊆ I of A such that x^k ∈ F for every x ∈ I. The ring A is said to be nonnil-SFT, if each nonnil-ideal (i.e., not contained in the nilradical of A) is SFT. We investigate the nonnil-SFT variant of some well known theorems on SFT-rings. Also we study the transfer of this property to Nagata's idealization and the amalgamation algebra along an ideal. Many examples are given. In fact, using the amalgamation construction, we give an infinite family of nonnil-SFT rings which are not SFT.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1349-1357
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• 2022

Let 𝒜 = (A_n)_n≥0 be an increasing sequence of rings. We say that 𝓘 = (I_n)_n≥0 is an associated sequence of ideals of 𝒜 if I₀ = A₀ and for each n ≥ 1, I_n is an ideal of A_n contained in I_n+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.

• Communications of the Korean Mathematical Society
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• v.39 no.3
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• pp.611-622
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• 2024

The purpose of this paper is to introduce a new class of rings containing the class of m-formally Noetherian rings and contained in the class of nonnil-SFT rings introduced and investigated by Benhissi and Dabbabi in 2023 [4]. Let A be a commutative ring with a unit. The ring A is said to be nonnil-m-formally Noetherian, where m ≥ 1 is an integer, if for each increasing sequence of nonnil ideals (I_n)_n≥0 of A the (increasing) sequence (∑_i1+⋯+im=nI_i1I_i2⋯I_im)_n≥0 is stationnary. We investigate the nonnil-m-formally Noetherian variant of some well known theorems on Noetherian and m-formally Noetherian rings. Also we study the transfer of this property to the trivial extension and the amalgamation algebra along an ideal. Among other results, it is shown that A is a nonnil-m-formally Noetherian ring if and only if the m-power of each nonnil radical ideal is finitely generated. Also, we prove that a flat overring of a nonnil-m-formally Noetherian ring is a nonnil-m-formally Noetherian. In addition, several characterizations are given. We establish some other results concerning m-formally Noetherian rings.

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.447-459
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• 2016

Let D be an integral domain, {$X_{\alpha}$} be a nonempty set of indeterminates over D, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}_1}$ be the first type power series ring over D. We show that if D is a t-SFT $Pr{\ddot{u}}fer$ v-multiplication domain, then $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_{1_{D-\{0\}}}$ is a Krull domain, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_1$ is a $Pr{\ddot{u}}fer$ v-multiplication domain if and only if D is a Krull domain.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1007-1012
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• 2018

Let R be an integral domain. We prove that the power series ring R[[X]] is a Krull domain if and only if R[[X]] is a generalized Krull domain and t-dim $R{\leq}1$, which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts "Krull domain" and "generalized Krull domain" are the same in power series rings, (2) there exists a non-t-SFT domain R with t-dim R > 1 such that t-dim R[[X]] = 1.

• Proceedings of the Korean Vacuum Society Conference
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• 2011.08a
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• pp.216-216
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• 2011

We will investigate the bonding configurations of phenylalanine and tyrosine adsorbed on the Ge(100) surface using CLPES and DFT calculations. First, the C 1s, N 1s, and O 1s spectra obtained at 300 K revealed that both the amine and carboxyl groups of phenylalanine and tyrosine concurrently participated in adsorption on the Ge(100) surface without bond breaking using CLPES, depending on the extent of coverage. In the second place, we confirmed that the "O-H dissociated-N dative bonded structure" is the most stable structure implying kinetically favorable structure, and the "O-H dissociation bonded structure" is another stable structure manifesting thermodynamically advantageous structure using DFT calculations. This tendency turns up both phenylalanine and tyrosine, similarly. Furthermore, through the CLPES data and DFT calculation data, we discovered that the "O-H dissociated-N dative bonded structure" and the "O-H dissociation bonded structure" are preferred at 0.30 ML and 0.60 ML, respectively. Moreover, we found that the phenyl ring of phenylalanine is located in axial position to Ge(100) surface, but the phenyl ring of tyrosine is located in parallel to Ge(100) surface using DFT calculations.