Definition and basic properties of modules over a ring of exchange

Introduction

In algebra, modules over rings are fundamental structures that generalize vector spaces over fields. A ring of exchange is a commutative ring with unity where the multiplication operation is associative and satisfies the properties of a ring. This article focuses on the definition and basic properties of modules over such rings, emphasizing their structural characteristics and applications. Modules over rings of exchange inherit many properties from their underlying rings, but their unique commutativity and unitarity introduce additional constraints and behaviors. The study of these modules is crucial for understanding homological algebra, representation theory, and the classification of algebraic structures.

Definitions

A module over a ring of exchange is a mathematical structure consisting of a set $ M $ equipped with two operations: addition and scalar multiplication by elements of the ring. These operations satisfy the following axioms:

  1. Addition is associative and commutative, and there exists a zero element $ 0 \in M $ such that $ m + 0 = 0 + m = m $ for all $ m \in M $.
  2. Scalar multiplication is compatible with addition: $ \lambda (m + n) = \lambda m + \lambda n $ for all $ \lambda \in \mathbb{R} $ (or the ring's scalars) and $ m, n \in M $.
  3. The scalar multiplication is distributive over addition: $ (\lambda + \mu) m = \lambda m + \mu m $ and $ \lambda (m + n) = \lambda m + \lambda n $.
  4. The scalar multiplication is associative with respect to the ring's multiplication: $ (\lambda \mu) m = \lambda (\mu m) $ for all $ \lambda, \mu \in \mathbb{R} $ and $ m \in M $.
  5. The ring's unity element $ 1 $ acts as the identity in scalar multiplication: $ 1 \cdot m = m $ for all $ m \in M $.

A module over a ring of exchange is said to be finitely generated if there exists a finite set of elements in $ M $ such that every element of $ M $ can be expressed as a linear combination of these elements with coefficients in the ring. Similarly, a module is free if it is isomorphic to a direct sum of copies of the ring, and it is projective if it is a direct summand of a free module.

Basic Properties

Modules over rings of exchange exhibit several key properties that distinguish them from general modules over arbitrary rings.

Finiteness and Submodules

A ring of exchange is finite if it has a finite number of elements, and its modules are finitely generated if they are generated by a finite set. This finiteness ensures that modules over such rings are finitely generated and projective if they are free. For example, in a finite ring $ R $, every module is finitely generated and projective if and only if it is free.

A submodule of a module $ M $ over a ring $ R $ is a subset $ N \subseteq M $ that is closed under addition and scalar multiplication. The quotient module $ M / N $ is defined as $ M $ with $ N $ identified with the zero element. If $ N $ is a submodule of $ M $, then $ M / N $ is a module over $ R $, and the quotient map $ \pi: M \to M / N $ is a ring homomorphism.

Homomorphisms and Isomorphism Theorems

Homomorphisms between modules over a ring of exchange are determined by the ring's structure. A ring homomorphism $ f