# Examples of finite dimensional algebras which do not satisfy the derived Jordan--H\"older property

## TL;DR

The paper constructs specific finite dimensional algebras that serve as counterexamples to the derived Jordan--H"older property, expanding understanding of algebraic structures where DJHP fails.

## Contribution

It introduces a method to build finite dimensional algebras with finite global dimension that do not satisfy DJHP, providing new counterexamples in algebra.

## Key findings

- Constructed matrix algebras from two elementary algebras
- Identified conditions under which DJHP fails for these algebras
- Provided examples of finite global dimension algebras lacking DJHP

## Abstract

We construct a matrix algebra $\Lambda(A,B)$ from two given finite dimensional elementary algebras $A$ and $B$ and give some sufficient conditions on $A$ and $B$ under which the derived Jordan--H\"older property (DJHP) fails for $\Lambda(A,B)$. This provides finite dimensional algebras of finite global dimension which do not satisfy DJHP.