# Decomposition theory of modules: the case of Kronecker algebra

## TL;DR

This paper introduces a simplified formula for decomposing finite-dimensional modules over algebras, specifically applied to the Kronecker algebra, facilitating computational approaches without explicit decomposition.

## Contribution

It provides a new, simpler formula for indecomposable decomposition of modules over finite-dimensional algebras, especially applied to Kronecker algebra, improving on previous methods.

## Key findings

- Simplified formula for module decomposition
- Explicit decomposition formula for Kronecker algebra
- Enables computer implementation for module analysis

## Abstract

Let $A$ be a finite-dimensional algebra over an algebraically closed field $\Bbbk$. For any finite-dimensional $A$-module $M$ we give a general formula that computes the indecomposable decomposition of $M$ without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mr\'oz. As an example we apply this formula to the Kronecker algebra $A$ and give an explicit formula to compute the indecomposable decomposition of $M$, which enables us to make a computer program.