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In this paper, we study the Cohen-Macaulayness of non-affine normalsemigroups in $\mathbb{Z}^n$. We do this by establishing the following fourstatements each of independent interest: 1) a Lazard type result on$I$-supported elements of $\prod_{\mathbb{N}}\mathbb{Q}_{\geq0}$ for an indexset $I\subset\mathbb{N}$; 2) a criterion of regularity of sequences of elementsof the ring via projective dimension; 3) a direct limit of polynomial ringswith toric maps; 4) any direct summand of rings of the third item isCohen-Macaulay. To illustrate the idea, we give many examples.

Cohen-Macaulayness of non-affine normal semigroups