$$K_0$$ K 0 -theory of n-potents in rings and algebras

Abstract

Let $$n \geqslant 2$$ be an integer. An n-potent is an element e of a ring R such that $$e^n = e$$ . 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 $$n \geqslant 2$$ . 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 n-homomorphism.