Modules and the Structure of Rings

A Primer

Golan

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351430364
• Edition: 1
• Topic: Mathematik
• Subtopic: Mathematik Allgemein

This textbook is designed for students with at least one solid semester of abstract algebra, some linear algebra background, and no previous knowledge of module theory. Modulesand the Structure of Rings details the use of modules over a ring as a means of consideringthe structure of the ring itself--explaining the mathematics and "inductivereasoning" used in working on ring theory challenges and emphasizing modules insteadof rings.Stressing the inductive aspect of mathematical research underlying the formal deductivestyle of the literature, this volume offers vital background on current methods for solvinghard classification problems of algebraic structures. Written in an informal butcompletely rigorous style, Modules and the Structure of Rings clarifies sophisticatedproofs... avoids the formalism of category theory... aids independent study or seminarwork... and supplies end-of-chapter problems.This book serves as an excellent primary.text for upper-level undergraduate and graduatestudents in one-semester courses on ring or module theory-laying a foundation formore advanced study of homological algebra or module theory.

Part One

PROJECTIVITY

2

Coproducts of Copies of R (Free Modules)

§1 INTRODUCTION

In Chapter 1 we examined the ring R regarded as a module over itself. We made an introductory survey of the summands, submodules, and factor modules of R. Through the use of the identity element of R, we encountered the concept of a projective module. The pattern of development of Chapter 1 and the concepts introduced there prove an introduction to the general theory of projectivity. The role played by the R-module R in Chapter 1 will now be taken over by the class of all coproducts of copies of the module R or, more generally, by the free left R-moduIes.

DEFINITION. An R-module is free if it is isomorphic to a coproduct of copies of the module R.

Chapter 2 is concerned with the free modules themselves. After it, we will take up successively the direct summands of free modules, the submodules of free modules, and the factor modules of free modules. In the course of extending our attention from free modules to these broader classes, we obtain a thorough introduction to the concept of projectivity. We will return to deeper structural investigations of projective modules in §5 of Chapter 9.

§2 BASES

The definition we have chosen for the concept of a free module is itself a structural definition of these modules. However, since free modules play such a fundamental role in module theory, we will also reformulate this definition in terms of sets of elements, and we will practice using this alternate, element-wise description.

For an arbitrary ring R and an arbitrary set Ω, let {R, I i ∈ Ω} be a family of copies of the R-module R indexed by the set Ω. Thus R, = R for each i in Ω. Let F = II{R_i, | i ∈ Ω}. Then the set F consists of those functions f from Ω to R satisfying the condition that f(i) = 0 for all but finitely-many elements i of Ω. With each i in Ω we associate the element b_i, of F defined by the conditions that b_i(j) = 1 if j equals i and b_i(j) = 0 otherwise. Let B = {b_i | i ∈ Ω}. Now suppose that f is an arbitrary element of F and, for each i in Ω, let r_i = f(i). Then r_i = 0 for all but a finite number of i in Ω and consequently the sum Σ{r_ib_i | i ∈ Ω} is defined and is an element of F. Since, for each j in Ω, we have (Σr_ib_i)(j) = Σr_ib_i)(j) = r_j, we see that f = Σr_ib_i. This B is a set of generators of F. But B has a further property. Suppose that {s_i | i ∈ Ω} is any indexed set of elements of R satisfying the condition that S_i = 0 for all but a finite number of i in Ω and that f = Σ_sibi. Then for each j in Ω we have f(j) = (Σ_sibi)(j) = Σ_sibi(j) = s_j, and consequently s_j = r_j for every j in Ω. This uniqueness of the representation of f as a linear combination of the elements of B shows that B is a basis for F in the following sense.

DEFINITION. A subset B of a left R-module A is a basis for A if for each element a of A there is a unique indexed set {r_b | b ∈ B} of elements of R for which r_b = 0 for all but a finite number of b in B and a = Σ{r_bb | b ∈B}.

Vector spaces over fields always possess bases but this is not the case for modules over arbitrary rings, even relatively nice ones. For example, the set ℤ/(3) of integers modulo 3, considered as a left ℤ-module, clearly has no basis. The possession of a basis is invariant under isomorphism. If α:A → A’ is an isomorphism of left R-modules and if B is a basis of A, then Bα = {bα | b ∈ B} is a basis of A’. It therefore follows that every free left R-module has a basis. We will show that the possession of a basis characterizes the free modules. For this purpose, the following powerful map-building tool will be convenient.

OBSERVATION 1. Let A be a left R-module having a basis B, and let A’ be an arbitrary left R-module. For each function g:B → A’ there is a unique map α:A → A’ satisfying bα = g(b) for all b in B.

VERIFICATION: Each element a of A can be written uniquely in the form Σ{r_bb | b ∈ B}, where the rb are elements of R. If there is a map α:A → A’ which agrees with g on B, then it must satisfy aα = (Σr_bb)α = Σr_bg(b). This shows that there is at most one map α:A → A’ which agrees with g on B. It also tells us that to complete the verification of the observation we need only consider the function α from A to A’ which sends a to Σr_bg(b) and verify that it is a map and that it agrees with g on B. For a = Σr_bb ∈ A and r ∈ R we have (ra)α = (Σrr_bb)α = Σrr_bg(b) = r[Σr_bg(b)] = r(aα). For a = Σr_bb and A’ = Σs_bb we have (a + A’)α = [Σ(r_b+s_b)b]α = Σ(r_b+s_b)g(b) = Σr_bg(b) + Σs_bg(b) = aα + A’α. Finally, for an element b′ of B the representation b′ = Σr_bb is simply the one given by r_b′ = 1 and r_b = 0 if b ≠ b′. Thus b′α = Σr_bg(b) = r_b'g(b′) = g(b′).

From the uniqueness condition in Observation 1 we see that, if A is a left R-module with basis B, to verify that maps α:A → A’ and β:A → A’ are identical we need only show that they agree on B. It is now convenient to observe the following basic characterization of free modules.

OBSERVATION 2. A left R-module is free if and only if it contains a basis.

VERIFICATION: We have already seen that a free module must have a basis; suppose that A is a left R-module having a basis I. Use I to index a family {R_i | i ∈ I} of copies of R and form F = ∐{R_i | i ∈ I}. Let B = {b_i | i ∈ I} be the basis of F as constructed in the initial discussion of this section. Let g:I → B be the bijection given by g(i) = b_i for all i in I, and let g’ be the inverse of g. There is a unique map α:A → F that agrees with g on I, and there is a unique map (α’:F → A that agrees with g’ on B. Then αα’ is a map from A to A that agrees with the identity function g’g on I and α’α is a map from F to F that agrees with the identity function gg’ on B. It follows that αα’ and α'α are identity maps and that α and α' are a pair of inverse isomorphisms.

This observation suggests that a routine way of showing that a module is free is to construct a basis of the module.

A free module may possess more than one basis. For the ℤ-moduIe ℤ, {1} is a basis and so is {−1}. For ℤ ∐ ℤ, the pair of elements {(1,0), (n, 1)} is a basis for any n in ℤ. Must two bases of the same free module have the same cardinality? The answer may be surprising. If the module has an infinite basis, order prevails: if F is a free module having an infinite basis, then any two bases of F must have the same cardinality. This fact is a consequence of the following observation, which contains extra information as well.

OBSERVATION 3. If F is a free left R-module having a basis B of infinite cardinality k then each subset A of F having cardinality h less than k is contained in a proper direct summand of F.

VERIFICATION: Each element a of A has a unique representation of the form a = Σ_b∈Br_bb with the elements r_b from R. Let B(a) = {b ∈ B | r_b ≠ 0}. Then each B(a) is finite and the cardinality of the union G = ∪{B(a) | a ∈ A} is finite if A is finite and is equal to h if A is infinite. In either case, it is less than k and we have F = [⊕{Rb | b ∈ C}] ⊕ [⊕{Rb I b ∈ B \ C}] and A⊆ ⊕{Rb | b ∈ C} ≠ F.

If a free module does not have an infinite basis, chaos may reign. Among the exercises, you are asked to show that there exists a ring R for which for every positive integer n there is a basis of the left R-module R that has precisely n elements. This, however, can happen only if the ring R is noncommutative. In Section §4 we will see a class of rings R such that if F is a free left R-module having a finite basis then all bases of F have the same cardinality. At later points in this book, we will show that for rings of particular types, any two bases of the same free module must have the same cardinality.

We will now make an important observation about free modules.

OBSERVATION 4. Free modules are projective.

VERIFICATION: Let F be a free left R-module and let α:A→A’ be an arbitrary epimorphism of left R-modules. If β:F → A’ is a map then we must construct a map β’:F → A satisfying β’α = β. We begin by choosing a basis B of F. Since α is surjective, for each b in B there exists there exists an element a_b of A for which a_bα = bβ. Define a function g:B → A by g(b) = a_b. Then there is a unique map β’:F → A that agrees with g on B. For each b in B, bβ′α = (bβ′)α = (g(b))α. = a_bα = bα. Thus β′α and β agree on a basis of F and so we conclude that β′α. = β, as required.

We have been explicit in mentioning the need for the Axiom of Choice when it arises and so at this point we should point out that Observation 4 is another statement equivalent to the axiom of choice. In fact, Blass [1979] has shown that the statement “every free ℤ-module is projective” is equivalent, given the Zermelo-Fraenkel axioms for set theory, to the Axiom of Choice.

It must be apparent that the specification of a basis of a free R-module F is very closely related to the specification of an isomorphis...