• 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.

• 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.

• Communications of the Korean Mathematical Society
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• v.39 no.3
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• pp.623-642
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• 2024

This paper introduces and studies a generalization of (n, d)-rings introduced and studied by Costa in 1994 to rings with prime nilradical. Among other things, we establish that the ϕ-von Neumann regular rings are exactly either ϕ-(0, 0) or ϕ-(1, 0) rings and that the ϕ-Prüfer rings which are strongly ϕ-rings are the ϕ-(1, 1) rings. We then introduce a new class of rings generalizing the class of n-coherent rings to characterize the nonnil-coherent rings introduced and studied by Bacem and Benhissi.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1095-1106
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• 2020

The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let 𝓗 = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. Let R ∈ 𝓗 be a ring with total quotient ring T(R) and define 𝜙 : T(R) → R_Nil(R) by ${\phi}({\frac{a}{b}})={\frac{a}{b}}$ for any a ∈ R and any regular element b of R. Then 𝜙 is a ring homomorphism from T(R) into R_Nil(R) and 𝜙 restricted to R is also a ring homomorphism from R into R_Nil(R) given by ${\phi}(x)={\frac{x}{1}}$ for every x ∈ R. We say that R is a 𝜙-pseudo-Krull ring if 𝜙(R) = ∩ R_i, where each R_i is a nonnil-Noetherian 𝜙-pseudo valuation overring of 𝜙(R) and for every non-nilpotent element x ∈ R, 𝜙(x) is a unit in all but finitely many R_i. We show that the theories of 𝜙-pseudo Krull rings resemble those of pseudo-Krull domains.