Foundations of Module and Ring Theory

Robert Wisbauer

• Publisher: CRC Press
• Year: 2018
• Print ISBN:
  9780367825386
• eBook ISBN:
  9781351447348
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Chapter 1

Elementary properties of rings

Before we deal with deeper results on the structure of rings with the help of module theory we want to provide elementary definitions and constructions in this chapter.

1 Basic notions

A ring is defined as a non-empty set R with two compositions +, ·: R × R → R with the properties:

(i)	(R, +) is an abelian group (zero element 0);
(ii)	(R, ·) is a semigroup;
(iii)	for all a,b,c ∈ R the distributivity laws are valid: $(a+b)c=ac+bc,a(b+c)=ab+ac$.

The ring R is called commutative if (R, ·) is a commutative semigroup, i.e. if ab = ba for all a, b ∈ R. In case the composition · is not necessarily associative we will talk about a non-associative ring.

An element e ∈ R is a left unit if ea = a for all a ∈ R. Similarly a right unit is defined. An element which is both a left and right unit is called a unit (also unity, identity) of R.

In the sequel R will always denote a ring. In this chapter we will not generally demand the existence of a unit in R but assume R ≠ {0}.

The symbol 0 will also denote the subset {0} ⊂ R.

1.1 For non-empty subsets A,B ⊂ R we define:

$\begin{array}{l}A+B:=\left\{a+b|a\in A,b\in B\right\}\subset R,\\ AB:=\left\{{\displaystyle {\sum }_{i\le k}{a}_i{b}_i|a_i\in A,b_i\in B,k\in \mathit{\mathbb{N}}}\right\}\subset R.\end{array}$

With these definitions we are also able to form the sum and product of finitely many non-empty subsets A,B,C,… of R. The following rules are easy to verify:

$(A+B)+C=A+(B+C),(AB)C=A(BC).$

It should be pointed out that (A + B)C = AC + BC is not always true. However, equality holds if 0 ∈ A ∩ B. For an arbitrary collection {A_λ}_Λ of subsets A_λ ⊂ R with 0 ∈ A_λ, Λ an index set, we can form a ’sum’:

${\displaystyle \sum _{\text{λ}\in \Lambda }{A}_{\text{λ}}:=\left\{{\displaystyle \sum a_{\text{λ}}|a_{\text{λ}}\in A_{\text{λ}},a_{\text{λ}}\ne 0\text{for}\text{only}\text{finitely}\text{many}\text{λ}\in \Lambda }\right\}}.$

A subgroup I of (R, +) is called a left ideal of R if RI ⊂ I, and a right ideal if IR ⊂ I. I is an ideal if it is both a left and right ideal.

I is a subring if II ⊂ I. Of course, every left or right ideal in R is also a subring of R. The intersection of (arbitrary many) (left, right) ideals is again a (left, right) ideal.

The following assertions for subsets A, B, C of R are easily verified:

If A is a left ideal, then AB is a left ideal.

If A is a left ideal and B is a right ideal, then AB is an ideal and BA ⊂ A ∩ B.

If A, B are (left, right...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Symbols
• Chapter 1 Elementary properties of rings
• Chapter 2 Module categories
• Chapter 3 Modules characterized by the Hom-functor
• Chapter 4 Notions derived from simple modules
• Chapter 5 Finiteness conditions in modules
• Chapter 6 Dual finiteness conditions
• Chapter 7 Pure sequences and derived notions
• Chapter 8 Modules described by means of projectivity
• Chapter 9 Relations between functors
• Chapter 10 Functor rings
• Bibliography
• Index