Subrings in trigonometric polynomial rings. (English) Zbl 1224.13003
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 \(x=x'x''\).
MSC:
13A05 | Divisibility and factorizations in commutative rings |
13B25 | Polynomials over commutative rings |
13B30 | Rings of fractions and localization for commutative rings |
12D05 | Polynomials in real and complex fields: factorization |
42A05 | Trigonometric polynomials, inequalities, extremal problems |