Embedding truncated skew polynomial rings into matrix rings and embedding of a ring into 2x2 supermatrices

TL;DR

This paper demonstrates how certain algebraic structures, like truncated skew polynomial rings and rings with involution, can be embedded into matrix rings, expanding understanding of their algebraic representations.

Contribution

It provides explicit embeddings of truncated skew polynomial rings and rings with involution into matrix rings, offering new tools for algebraic analysis.

Findings

Truncated skew polynomial rings embed into matrix rings under certain conditions.

Rings with involution can be embedded into supermatrix algebras.

These embeddings facilitate algebraic computations and structural analysis.

Abstract

For an endomorphism s of R with s^{t}=1 we prove that the truncated polynomial ring (algebra) R[w,s]/(w^{t}) embeds into M_{t}(R[z]/(z^{t})). For an involution we exhibit an embedding of R into M_{2,1}^{s}(R), where M_{2,1}^{s}(R) is the algebra of the so called (s,2,1) supermatrices.