• Kyungpook Mathematical Journal
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• v.50 no.1
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• pp.1-5
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• 2010

Let P be a prime ideal of a commutative unital ring R; X an indeterminate; D := R/P; L the quotient field of D; F an algebraic closure of L; ${\alpha}$ ${\in}$ L[X] a monic irreducible polynomial; ${\xi}$ any root of in F; and Q = ${\alpha}$>, the upper to P with respect to ${\alpha}$. Then R[X]/Q is R-algebra isomorphic to $D[{\xi}]$; and is R-isomorphic to an overring of D if and only if deg(${\alpha}$) = 1.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.453-468
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• 2022

In this article, we define a module M to be G^z-extending if and only if for each z-closed submodule X of M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We investigate structural properties of G^z-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for G^z-extending modules. We obtain that if a ring is right G^z-extending, then so is its essential overring. Also it is shown that the G^z-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right G^z-extending ring.

• 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.

• 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.