# Centrally Stable Algebras

## TL;DR

This paper introduces the concept of centrally stable algebras, characterizes their structure over perfect fields, and explores their behavior in tensor products, providing a comprehensive classification of finite-dimensional cases.

## Contribution

It defines centrally stable algebras, analyzes their properties, especially in tensor products, and characterizes finite-dimensional centrally stable algebras over perfect fields.

## Key findings

- Centrally stable algebras are characterized by their centers under epimorphisms.
- Finite-dimensional centrally stable algebras over perfect fields are classified as direct products of tensor products involving central simple algebras.
- The main theorem provides a structural decomposition for these algebras.

## Abstract

We define an algebra $A$ to be centrally stable if, for every epimorhism $\varphi$ from $A$ to another algebra $B$, the center $Z(B)$ of $B$ is equal to $\varphi(Z(A))$, the image of the center of $A$. After providing some examples and basic observations, we consider in somewhat greater detail central stability in tensor products of algebras, and finally establish our main result which states that a finite-dimensional unital algebra $A$ over a perfect field $F$ is centrally stable if and only if $A$ is isomorphic to a direct product of algebras of the form $C_i\otimes_{F_i}A_i$, where $F_i$ is a field extension of $F$, $C_i$ is a commutative $F_i$-algebra, and $A_i$ is a central simple $F_i$-algebra.