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#### Groups, Rings, Modules

**By Maurice Auslander, David A. Buchsbaum**

**Dover Publications, Inc.**

**Copyright © 1974 Maurice Auslander and David A. Buchsbaum**

All rights reserved.

ISBN: 978-0-486-79542-3

All rights reserved.

ISBN: 978-0-486-79542-3

CHAPTER 1

**SETS AND MAPS**

**INTRODUCTION**

This chapter and the next are devoted to a review of the basic concepts of set and group theory. Because we are assuming the reader already has some familiarity with these topics, our exposition is neither systematic nor complete. Only a brief description of the basic concepts and results that are needed in the rest of this book is presented.

This should serve to give the reader some idea of the mathematical background we are assuming as well as help fix conventions and notations for the rest of the book. Although few proofs are given, outlines of proofs of the less obvious results cited in the text are given in the exercises. It is hoped that the reader will find completing these outlines a useful way of familiarizing himself with any new concepts or results he may encounter in this or the next chapter.

**1. SETS AND SUBSETS**

We take a naive, nonaxiomatic view of set theory. We view a set as an actual collection of things called the elements of the set. We will often denote the fact that *x* is an element of the set *X* by writing *x* [member of] *X.* From this point of view it is obvious that two sets are the same if and only if they have the same elements. Or stated more precisely, two sets *X* and *Y* are the same if and only if both of the following statements are true:

(a) If *x* [member of] *X,* then *x* [member of] *Y.*

(b) If *y* [member of] *Y,* then *y* [member of] *X.*

In this connection, we remind the reader that in mathematical usage, a statement of the form "If *A,* then *B*" is true unless *A* is true and *B* is false, in which case it is false. In particular, if *A* is false, then the statement "If *A,* then *B*" is true independent of whether *B* is true or false. To illustrate this point we show that there is only one empty set.

We recall that a set *X* is said to be empty if *X* has no elements; or more precisely, if the statement "*x* [member of] *X*" is always false. Suppose now that the sets *X* and *Y* are empty. Then both of the statements "*x* [member of] *X*" and "*y* [member of] *Y*" are always false. Hence, by our convention concerning sentences of the form "If *A,* then *B,*" both of the statements

(a) If *x* [member of] *X,* then *x* [member of] Y;

(b) If *y* [member of] *Y,* then *y* [member of] *X;*

are true. This shows that if the sets *X* and *Y* are both empty, then *X* = Y. Following the usual conventions of set theory, we assume that there is an empty set. This uniquely determined set will usually be denoted by 0.

An important set associated with a set *X* is the **power set 2***x* of *X* which we will define once we have recalled the notion of a subset of a set.

A set *Y* is said to be a subset of a set *X* if every element of *Y* is also an element of *X,* or equivalently, the set *Y* is a subset of the set *X* if and only if the statement "If *y**Y,* then *y* [member of] *X*" is true. The fact that *Y* is a subset of *X* is often denoted by *Y* [subset] *X,* which is sometimes also read as "*Y* is contained in *X.*"

One easily verified consequence of this definition is that if *X* is any set, then the empty set 0 is a subset of *X.* For the statement "If *x*[member of] 0, then *x* [member of] *X,*" is true for any set *X* because the statement "*x*[member of] 0" is always false. Also associated with an element *x* of *X* is the subset {*x*} of *X* consisting precisely of the element *x* of *X.* Further, the reader should have no difficulty verifying the following.

**Basic Properties 1.1**

Let *X, Y,* and *Z* be sets. Then:

(a) *X* [subset] *X.*

(b) *X = Y* if and only if *X* [subset] *Y* and *Y* [subset] *X.*

(c) If *X* [subset] *Y* and *Y* [subset] *Z,* then *X* [subset] *Z.*

We are now in a position to define the power set 2*x* of a set *X.* The set 2*x* is the set whose elements are precisely the subsets of *X.* Stated symbolically, the power set 2*x* of a set *X* is the set with the property that *Y* [member of] 2*x* if and only if *Y* [subset] *X.*

It is worth noting that 2*x* is never empty, even if *X* is empty. This is because the empty set is always contained in *X* and is thus an element of 2*x*. Also, as we have already observed, there is associated with each element *x* of *X* the element {*x*} of 2*x*. Hence, 2*x* consists of a single element if and only if *X* is empty.

We now recall the familiar notions of union and intersection of sets. Suppose *X* is a set and L a subset of 2*x*. The **intersection** of the subsets of *X* in L is the subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of *X* consisting of all *x* in *X* such that the statement "If *X'*[member of] L, then *x* [member of] *X*'" is true. It should be noted that if the subset L of 2*x* is empty, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For if L is empty, then the statement "If *X'*[member of] L, then *x* [member of] *X'*" is true for all *x* in *X* since the statement *"X'*[member of] L is false.

The **union** of the subsets of *X* in L is the subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of *X* consisting of all *x* in *X* with the property that the statement "There is an *X'*[member of] L such that *x* [member of] *X'"* is true. It should be noted that if L is empty, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For if L is empty, then the statement "There is an *X'*[member of] L such that *x* [member of] *X'"* is false for all *x* [member of] *X* since there are no *X'* in 2*x* satisfying the condition that *X'* is in L.

In practice, a particularly useful way of studying a set is to represent it as a union of some of its subsets. For this reason it is convenient to make the following definition.

**Definition**

Suppose *X* is a set. A subset L of 2*x* is called a **covering of X** if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Although coverings of various types play an important role in all of mathematics, we will be particularly concerned with the type of coverings called partitions.

**Definition**

A covering L of a set *X* is said to be a **partition of X** provided:

(a) If *X'*[member of] L, then *X'* ≠ 0.

(b) If *X'* and *X"* are distinct elements of L, then *X'* [intersection] *X"* = 0.

The reader should convince himself that a set of nonempty subsets of a set *X* is a partition of *X* if and only if each element in *X* is in one and only one subset of *X* in L. For this reason, if L is a partition of a set *X,* it makes sense to talk about the element of L containing a particular element *x* of *X.* We will usually denote by {x}L the unique element of the partition L of *X* containing the element *x* of *X.* When there is no danger of ambiguity concerning the particular partition L of a set *X,* we will write [x] for {x}L.

Finally, we recall what is meant by the **product***X* × *Y* of two sets *X* and *Y.* The set *X* × *Y* consists of all symbols (*x, y*) with *x* an element of *X* and *y* an element of *Y.* Hence, two elements (*x, y*) and (x', y') in *X* × *Y* are the same if and only if *x* = X' and *y* = y'. Obviously, *X* × *Y* is empty if and only if either *X* or *Y* is empty.

**2. MAPS**

A **map of sets** consists of three things: a set *X* called the **domain** of the map, a set *Y* called the **range** of the map, and a subset *f* of *X* × *Y* having the property that if *x* is in *X,* then there is a unique *y* in *Y* such that the element (*x, y*) in *X* × *Y* is in *f.* These data *X, Y, f* will be denoted by *f: X -> Y* which is to be read as "f is a map from *X* to *Y.*" If *x* is in *X,* then the unique element *y* in *Y* such that (*x, y* ) is in *f* is called the **value of the map f at x** and is denoted by

*f(x*).

It is important to observe that according to this definition two maps cannot be the same unless they have the same domain and range. Also, two maps *f: X->Y* and *g : X->Y* with the same domains and ranges are the same if and only if their values are the same for each *x* in *X,* that is, if and only if *f* (*x*) = *g*(*x*) for all *x* in *X.* Thus, once having specified the domain and range of a map, it only remains to describe its values for each *x* in *X* in order to completely determine the map. In the future, when defining particular maps from a set *X* to a set *Y,* we shall generally describe them by prescribing their values for each *x* in *X* rather than by writing down a subset of *X* × *Y.* In following this procedure it is of course necessary to make sure that one and only one value in the range has been assigned to each element of the domain. As an illustration of this point suppose that L is a partition of a set *X.* Then we have already seen that for each *x* in *X* there is one and only one element {x}L of L containing *x.* Thus, we obtain a map kL: X -> L by setting kL(x) = {x}L. Of course, we could have also defined the map kL as the subset of *X × T* consisting of all elements (x, [x]L) in *X* × L with *x* in *X.*

We now describe some important maps of sets.

**Example 2.1** Suppose *f:X->Y* is a map and *X'* is a subset of *X.* We define a map *f*|*X' : X' -> Y* called the **restriction of f to X'** by (*f*|*X'*)(*X'*) = *f*(*x'*) for all x' in *X'*

**Example 2.2** Associated with each subset *X'* of a set *X* is the **inclusion map** from *X'* to *X* which is denoted by inc : *X'->X* and is defined by inc(*x*) = *x* if *x* is an element of *X* which is in *X'.*

**Example 2.3** The inclusion map of a set *X* to itself is called the **identity map** and is usually denoted by id*x* for each set *X.*

**Example 2.4** Since the empty set 0 is a subset of any set *X,* we always have the inclusion map inc : 0->*X.* Actually, this is the only map from 0 to *X* and this unique map from 0 to a set *X* is called the **empty map**. In this connection, the reader should convince himself that there are no maps from a nonempty set to the empty set.

**Example 2.5** We have already seen that associated with a partition L of a set *X* is the map kL given by kL(x) = [x] for each *x* in *X* where [*x*] is the unique subset of *X* in containing L the element *x.* This map kL: X -> L is called the **canonical** or **natural** map from the set *X* to the partition L.

Suppose *X* and *Y* are sets. Then each map with domain *X* and range *Y* is completely determined by a subset of *X* × *Y* and hence by an element of 2*X×Y* Thus, the collection of all maps from *X* to *Y* which we denote by (*X, Y*) is a set which is a subset of 2*X×Y*.

Of fundamental importance in constructing and analyzing maps is the notion of the **composition of maps**. Given two maps *f: X->Y* and *g: Y->Z* with the range of *f* the same as the domain of *g,* we define their composition *gf* to be the map *gf: X->Z* given by *gf* (*x*)= *g* (*f* (*x*)) for each *x* in *X.* It follows immediately from this definition that if we are given three maps *f: U>X, g: X->Y,* and *h: Y->Z,* then the two maps *h* (*gf*):*U->Z* and (*hg*)*f: U->Z* are the same. This property of the composition of maps is referred to as the **associativity of the composition of maps**.

As an example of the composition of maps we point out that if f: *X* ->*Y* is a map of sets and *X'* is a subset of *X,* then *f*|*X': X' ->0 Y,* the restriction of *f* to *X',* is the composition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where inc : *X'* ->*X* is the inclusion map.

**3. ISOMORPHISMS OF SETS**

One of the most important problems in mathematics is deciding when two mathematical objects have the same or similar mathematical properties and can therefore be considered essentially the same. Since all the mathematical objects we will be considering in this book consist of an underlying set together with some additional structure, it is reasonable to first consider how sets are compared and the circumstances under which they are considered essentially the same.

Because a map from a set *X* to a set *Y* associates with each element *x* in *X* an element *y* in *Y,* a map clearly can be viewed as a method for comparing the sets *X* and*Y.* If this is a reasonable idea, then we should be able to state in terms of maps what is probably the simplest comparison of sets we can make: the fact that a set is the same as itself. The reader should have no difficulty convincing himself that the identity map on a set does indeed express this fact. It is interesting to note that the identity map on a set can be completely described in terms of maps as is done in the following.

**Basic Property 3.1**

For a map *f: X -> X,* the following statements are equivalent:

(a) *f* = id*x*.

(b) Given any map *g: X* ->*Y,* then *gf = g.*

(c) Given any map *h: Y* ->*X,* then *fh = h.*

Having decided that the identity map expresses the fact that a set is the same as itself, it is reasonable to ask what kind of maps between two sets *X* and *Y* must exist in order to conclude that *X* and *Y* resemble each other as much as possible. In view of our previous discussion, this amounts to asking when is a map *f : X -> Y* close to being an identity map? A possible answer might be that there is a map *g: Y* ->*X* such that the composition *gf: X* ->*X* is the identity on *X.* But there is no reason to favor the set *X* over the set *Y.* Hence, we should also require that there be a map *h: Y* ->*X* such that *fh* = idY. However, the associativity of the composition of maps implies that under these circumstances the two maps *g* and *h* are the same. Therefore, it seems reasonable to consider two sets *X* and *Y* as being essentially the same if there exists a pair of maps *f: X -> Y* and *g: Y -> X* such that *gf* = id*X* and *fg* = id*Y*. In fact, this amounts to nothing more than the familiar notion of two sets being isomorphic, as we see in the following.

*(Continues...)*

Excerpted fromGroups, Rings, ModulesbyMaurice Auslander, David A. Buchsbaum. Copyright © 1974 Maurice Auslander and David A. Buchsbaum. Excerpted by permission of Dover Publications, Inc..

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