Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

Abstract

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ⊆ 𝒩(R) (𝒩(R) ⊆ 𝒜(R)) if and only if 𝒜(R/P) ⊆ 𝒩(R/P) (𝒩(R/P) ⊆ 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ⊆ 𝒩(R/P) (𝒩(R/P) ⊆ 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ⊆ 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.

Key Words:

• Amen rings
• Atomic modules
• Centrally prime ideals
• Critical modules
• Duo rings
• Finitely embedded modules
• Finite R-algebra
• Krull dimension
• Noetherian dimension
• Normal prime ideals
• Valuation rings

2000 Mathematics Subject Classification:

• Primary 16A53, 16P20
• Secondary 16A56

ACKNOWLEDGMENTS

The authors would like to thank the referee for a careful reading of this article. Thanks are also due to professor Albu for his useful comments.

Notes

Communicated by T. Albu.