Some Results on Triangular Coefficient Matrix Rings

TL;DR

This paper introduces the concept of triangular coefficient matrix rings, explores their ideal structure, characterizes radicals of certain polynomial rings, and examines property transfers between base and matrix rings.

Contribution

It defines triangular coefficient matrix rings and analyzes their ideal structure, radicals, and property transfer with base rings and subrings.

Findings

Characterization of radicals in Hurwitz polynomial rings

Structural insights into ideals of triangular coefficient matrix rings

Property transfer mechanisms between base rings and matrix subrings

Abstract

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.