Introduction

The basic nomenclature for modules and module homomorphisms is defined. Direct sums and products of modules are introduced. Split short exact sequences are discussed. Existence and universal properties of direct and inverse limits are established.

Direct limits generalize direct sums, inverse limits – direct products. This topic is covered in Chapter 26–6, but could very well be covered at the end of the present Chapter 1. The construction of direct and inverse limits of modules and rings is a good exercise in using all the concepts introduced in this Chapter 1. Furthermore, they are a rich source of nontrivial examples of modules and rings.

Definitions

Throughout, R is an arbitrary ring with or without an identity element.

Definition. An additive abelian group M with addition denoted by + is a right R-module if there is a function M × R → M, (m, r) → mr, for m ∈ M, r ∈ R, such that for any x, y ∈ M and any a, b ∈ R the following hold:

1. (i) (x + y)a = xa + yb, x(a + b) = xa + yb;

2. (ii) x(ab) = (xa)b.

Notation. The notation M = MR will mean that R is a ring and M is a right R-module (and similarly V= RV for left modules). The zero module will be denoted by either one of the three {0} = (0) = 0.

For the remainder of this section, M = MR.