  Let $A$ be a direct limit of a direct system of Cohen-Macaulay rings. In this
paper, we describe the Cohen-Macaulay property of $A$. Our results indicate
that $A$ is not necessarily Cohen-Macaulay. We show $A$ is Cohen-Macaulay under
various assumptions. As an application, we study Cohen-Macaulayness of
non-affine normal semigroup rings.
