Abstract
We denote by ๐(R) the class of all Artinian R-modules and by ๐ฉ(R) the class of all Noetherian R-modules. It is shown that ๐(R) โ ๐ฉ(R) (๐ฉ(R) โ ๐(R)) if and only if ๐(R/P) โ ๐ฉ(R/P) (๐ฉ(R/P) โ ๐(R/P)), for all centrally prime ideals P (i.e., ab โ P, a or b in the center of R, then a โ P or b โ P). Equivalently, if and only if ๐(R/P) โ ๐ฉ(R/P) (๐ฉ(R/P) โ ๐(R/P)) for all normal prime ideals P of R (i.e., ab โ P, a, b normalize R, then a โ P or b โ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, ๐(R) โ ๐ฉ(R) implies that ๐ฉ(R) = ๐(R), where R is a duo ring. For a ring R, we prove that ๐ฉ(R) = ๐(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 ฮฑ-atomic and ฮฒ-critical for some ordinals ฮฑ,ฮฒ โฅ 1 and in fact ฮฑ = ฮฒ = 1.
2000 Mathematics Subject Classification:
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.