We explore the subrings in trigonometric polynomial rings and their factorization properties. Consider the ring $S'$ of complex trigonometric polynomials over the field $\mathbb{Q}(i)$ (see \cite{SU}). We construct the subrings $S'_1$, $S'_0$ of $S'$ such that $S'_1\subseteq S'_0\subseteq S'$. Then $S'_1$ is a Euclidean domain, whereas $S'_0$ is a Noetherian HFD. We also characterize the irreducible elements of $S'_1$, $S'_0$ and discuss among these structures the condition: Let $A\subseteq B$ be a unitary (commutative) ring extension. For each $x\in B$ there exist $x'\in U(B)$ and $x''\in A$ such that $x=x'x''$.