From Thin Schemes to Modules

In this chapter, we shall develop some of the fundamental aspects of the representation theoretic part of scheme theory. Representations of (finite) schemes reflect the arithmetic structure of schemes. They are useful in cases where the structure constants underly extreme constraints. 

The central notion in representation theory of finite schemes is the one of an associative ring. Rings give rise to modules. 

Since rings as well as modules are built from commutative groups and groups are identified with thin schemes (via the group correspondence) one may view the notion of a ring and the one of a module as part of thin scheme theory . 

The first section of this chapter provides general observations on modules over associative rings with 1. The collection includes the Homomorphism Theorem and the Isomorphism Theorem for modules over associative rings with 1. 

In the second section, we shall look at commutative associative rings with 1. We prove that the set of all elements of a commutative associative ring D with 1 which are integral over a unitary subring of D forms a ring. 

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