The set of all right topologizing filters on a fixed but arbitrary ring R admits a monoid operation ‘’ that is in general noncommutative, even in cases where the ring R is commutative. Earlier results show (see [8]) that commutativity of the monoid operation ‘’, when imposed as a condition on manifests as a type of finiteness condition on R. In a quite separate and much earlier study, Shores [6] has shown that if R is a commutative semiartinian ring, then R will be artinian precisely if the first two terms in the Loewy series for RR, namely and are finitely generated. Shores goes further to produce examples which show that the finiteness of just exercises no constraint whatsoever on the length of RR. The main result of this paper asserts that a commutative semiartinian ring R will be artinian precisely if is finitely generated and the monoid operation ‘’ on is commutative. A family of commutative semiartinian rings of Loewy length 3 is constructed and this used to delineate earlier theory. In particular, and within this family, rings R are exhibited such that (1) and have infinite length, yet (2) the monoid operation ‘’ on is commutative.