K_0-theory of n-potents in rings and algebras

Abstract

Let $n\ge 2$ 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 $K_0^n(A)\cong (K_0(A){)}^{n-1}$ for all $n\ge 2$. 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}.

arXiv commentTo appear in the European Journal of Mathematics