Remarks on Almost Artinian Rings

In this note, all rings are assumed to be commutative with identity. Let R be a ring. We shall call a R-module almost Artinian if every proper homomorphic image is an Artinian R-module. The polynomial ring k[X] over a fieldk is an example of an almost Artinian k[X]-module which is not Artinian. Almost Artinian modules have been studied by Cohen [1], Matlis [5] and J.Hein [3]. The purpose of this note is to characterize the total quotient ring which is almost Artinian in terms of ringed spaces.