Modules whose hereditary pretorsion classes are closed under products

A module M is called product closed if every hereditary pretorsion class in σ[M ] is closed under products in σ[M ]. Every module which is locally of finite length is product closed and every product closed module is semilocal. LetM ∈ R-Mod be product closed and projective in σ[M ]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invariant submodules; (3) if M is finitely generated and every hereditary pretorsion class in σ[M ] is M -dominated, then M has finite length. If the ring R is commutative it is proven that M is product closed if and only if M is locally of finite length. An example is provided of a product closed module with zero socle. 1991 Mathematics Subject Classification: primary 16S90. It was shown by Beachy and Blair [2, Proposition 1.4, p. 7 and Corollary 3.3, p. 25] that the following three conditions on a ring R with identity are equivalent: (1) every hereditary pretorsion class in R-Mod is closed under arbitrary (and not just finite) direct products, or equivalently, every left topologizing filter on R is closed under arbitrary (and not just finite) intersections; (2) every left R-module M is finitely annihilated, meaning (0 :M) = (0 : X) for some finite subset X of M ; (3) R is left artinian.