Schemes of Associative Algebras

TL;DR

This paper introduces the concept of associative schemes, extending the classical scheme theory from commutative to associative rings using completions in simple modules.

Contribution

It provides a new framework for defining associative schemes over a field, generalizing the classical scheme theory to non-commutative rings.

Findings

Defined associative schemes over a field.

Extended scheme theory from commutative to associative rings.

Provided a method using completions in simple modules.

Abstract

We give a definition of associative schemes, schemes of associative rings, over a field $k,$ using the definition of completion of an associative $k$-algebra in a finite set of simple modules. We start by giving a weaker but sufficient definition of ordinary schemes of commutative rings which can be generalized to associative rings.