[Sigma]-algebraically compact modules and L [omega] 1 [omega]-compact cardinals

We prove that the property Add (M )Prod (M ) characterizes Σ-algebraically compact modules if |M | is not [omega]-measurable. Moreover, under a large cardinal assumption, we show that over any ring R where |R | is not [omega]-measurable, any free module M of [omega]-measurable rank satisfies Add (M )Prod (M ), hence the assumption on |M | cannot be dropped in general (e.g., over small non-right perfect rings). In this way, we extend results from a recent paper by Simion Breaz .