# Codes over an algebra over ring

## TL;DR

This paper explores the structure of linear codes over a specific algebraic ring constructed from a finite Frobenius ring and idempotent variables, expanding understanding of code properties in algebraic coding theory.

## Contribution

It introduces new structures of linear codes over the algebra $\\mathcal{R}_k$ built from a finite Frobenius ring and idempotent variables, providing foundational insights.

## Key findings

- Characterization of linear codes over the algebra \mathcal{R}_k
- Analysis of code properties over the ring \mathcal{R}_k
- Potential applications in coding theory and algebraic structures

## Abstract

In this paper, we consider some structures of linear codes over the ring $\mathcal{R}_k=R[v_1,\dots,v_k],$ where $v_i^2=v_i$ forall $i=1,\dots,k),$ and $R$ is a finite commutative Frobenius ring.