Some Results on Triangular Coefficient Matrix Rings

In this paper, we introduce the concept of a {\it triangular coefficient matrix ring} and investigate the structure of its ideals. We then characterize the radicals of the ring $R_{h}[x]/\langle x^{n} \rangle$ for every positive integer $n$, where $R_{h}[x]$ denotes the Hurwitz polynomial ring and $\langle x^{n} \rangle$ represents the ideal of this ring generated by $x^{n}$. Furthermore, we explore several properties that are transferred between the base ring $R$ and the matrix ring $H_{n}(R)$ which is a proper subring of the triangular coefficient matrix ring.