On a weak form of equational compactness

In this paper, we discuss a property of large systems of equations over universal algebras which does not appear to be generally known, but which coincides with equational compactness for abelian groups. It is shown, for example, that if α is a cardinal number with uncountable cofinality, then every finitely solvable system of α equations over any countable algebra has a solvable subsystem consisting also of α equations. As an application, this property is used to generalize some results of Jensen and Lenzing on the non-compactness of ultrapowers of modules.