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http://dx.doi.org/10.4134/CKMS.c200116

ON 𝜙-PSEUDO-KRULL RINGS  

El Khalfi, Abdelhaq (Modelling and Mathematical Structures Laboratory Department of Mathematics Faculty of Science and Technology of Fez)
Kim, Hwankoo (Division of Computer and Information Engineering Hoseo University)
Mahdou, Najib (Modelling and Mathematical Structures Laboratory Department of Mathematics Faculty of Science and Technology of Fez)
Publication Information
Communications of the Korean Mathematical Society / v.35, no.4, 2020 , pp. 1095-1106 More about this Journal
Abstract
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) → RNil(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 RNil(R) and 𝜙 restricted to R is also a ring homomorphism from R into RNil(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) = ∩ Ri, where each Ri 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 Ri. We show that the theories of 𝜙-pseudo Krull rings resemble those of pseudo-Krull domains.
Keywords
Amalgamated algebra; nonnil-Noetherian ring; pseudo-Krull domain; pseudo-valuation ring; ${\phi}$-pseudo-Krull ring; trivial ring extension;
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