Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2020.04.02
  • Accepted : 2020.07.02
  • Published : 2020.10.31
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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.

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References

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