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

• Published in: Communications in algebra 2009-02, Vol.37 (2), p.650-662
• Main Authors: Hashemi, J., Karamzadeh, O. A. S., Shirali, N.
• Format: Article
• Language: English
• Subjects: Amen rings; Atomic modules; Centrally prime ideals; Critical modules; Duo rings; Finite R-algebra; Finitely embedded modules; Krull dimension; Noetherian dimension; Normal prime ideals; Primary 16A53, 16P20; Secondary 16A56; Valuation rings
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927870802254835

We denote by (R) the class of all Artinian R-modules and by (R) the class of all Noetherian R-modules. It is shown that (R) ⊆  (R) ( (R) ⊆  (R)) if and only if (R/P) ⊆  (R/P) ( (R/P) ⊆  (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 (R/P) ⊆  (R/P) ( (R/P) ⊆  (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, (R) ⊆  (R) implies that (R) =  (R), where R is a duo ring. For a ring R, we prove that (R) =  (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.