Summary: We explore the subrings in trigonometric polynomial rings. Consider the rings \(T\) and \(T'\) of real and complex trigonometric polynomials over the fields \(\mathbb R\) and its algebraic extension \(\mathbb C\) respectively. We construct the subrings \(T_0\) of \(T\) and \(T'_0\), \(T_1'\) of \(T'\). Then \(T_0\) is a BFD whereas \(T_0'\) and \(T_1'\) are Euclidean domains. We also discuss among these rings 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 \(x=x'x''\).