On extensions of Dirichlet and Green–Tao theorems and Goldbach–Dirichlet representations over certain families of commutative rings with unity

In this paper, we study stronger forms of Goldbach's conjecture enriched with the linear representations of prime numbers given by the classical Dirichlet theorem and its extensions. We call such a representation a Goldbach–Dirichlet representation (GD‐representation). Among other results, we show that Dirichlet's theorem on arithmetic progressions is, in general, not true in the ring of formal power series over the integers. Additionally, we use a polynomial version of the Schinzel hypothesis due to A. Bodin, P. Dèbes, and S. Najib to prove the existence of GD‐representations for a wide collection of polynomial rings over special families of fields of characteristic zero, among others. Moreover, we study the (non)validity of Dirichlet's theorem over several families of commutative rings with unity like polynomial and formal series rings. Finally, we obtain a generalization for polyomial rings of the celebrated Green–Tao theorem.