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 $R_h[x]\mathrm{/}⟨x^n⟩$ for every positive integer $n$, where $R_h[x]$ denotes the Hurwitz polynomial ring and $⟨x^n⟩$ 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.