# K_0-theory of n-potents in rings and algebras

## TL;DR

This paper introduces a new K-theory group based on n-potents in rings and algebras, exploring its properties, isomorphisms with classical K-theory, and functorial behavior under generalized homomorphisms.

## Contribution

It defines and studies the K_0^n group for n-potents, establishing isomorphisms with classical K-theory for complex algebras and analyzing functorial properties.

## Key findings

- K_0^n(A) is isomorphic to (K_0(A))^{n-1} for complex algebras
- The isomorphism does not hold in general over cyclotomic fields
- K_0^n is functorial for n-homomorphisms

## Abstract

Let $n \geq 2$ be an integer. An \emph{$n$-potent} is an element $e$ of a ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a complex algebra, there is a group isomorphism $K_0^n(A) \cong \bigl(K_0(A)\bigr)^{n-1}$ for all $n \geq 2$. However, for algebras over cyclotomic fields, this is not true in general. We consider $K_0^n$ as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an \emph{$n$-homomorphism}.