Modules whose hereditary pretorsion classes are closed under products

A module MM is called product closed   if every hereditary pretorsion class in σ[M]σ[M] is closed under products in σ[M]σ[M]. Every module MM which is locally of finite length (every finitely generated submodule of MM has finite length) is product closed and every product closed module MM is semilocal (M/J(M)M/J(M) is semisimple). Let M∈R-Mod be product closed and projective in σ[M]σ[M]. It is shown that (1) MM is semiartinian; (2) if MM is finitely generated then MM satisfies the DCC on fully invariant submodules; (3) MM has finite length if MM is finitely generated and every hereditary pretorsion class in σ[M]σ[M] is MM-dominated. If the ring RR is commutative it is proven that MM is product closed if and only if MM is locally of finite length.