We denote by đ(R) the class of all Artinian R-modules and by đ©(R) the class of all Noetherian R-modules. It is shown that đ(R) â đ©(R) (đ©(R) â đ(R)) if and only if đ(R/P) â đ©(R/P) (đ©(R/P) â đ(R/P)), for all centrally prime ideals P (i.e., ab â P, a or b in the center of R, then a â P or b â P). Equivalently, if and only if đ(R/P) â đ©(R/P) (đ©(R/P) â đ(R/P)) for all normal prime ideals P of R (i.e., ab â P, a, b normalize R, then a â P or b â P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, đ(R) â đ©(R) implies that đ©(R) = đ(R), where R is a duo ring. For a ring R, we prove that đ©(R) = đ(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 ÎČ-critical for some ordinals α,ÎČ â„ 1 and in fact α = ÎČ = 1.