On a problem of Wenzel on equational compactness of modules

In this note, we introduce a weak form of equational compactness of modules, which we called x ₀-compactness, together with some of its basic properties. We then use it to settle in the negative a problem of G. H. Wenzel. We also prove the adjacent result that, over a wide class of commutative rings, if every module is x ₀-compact then the ring has finite representation type. Received April 3, 1997; accepted in final form October 1, 1997.