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OBJ (Application/Pdf) IDEALS AND RADICALS IN RING THEORY A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY CORNELL GRANT DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA DECEMBER, 1976 CONTENTS ACKNOWLEDGEMENTS ii Chapter I. INTRODUCTION 1 II. IDEALS IN RINGS 6 III. RADICALS IN RINGS 15 IV. IDEALS AND RADICALS IN RINGS WITH THE CHAIN CONDITIONS 22 V. APPLICATIONS 31 SELECTED BIBLIOGRAPHY 36 ACKNOWLEGDEMENT I wish to thank Dr. Johnny L. Houston for his gui¬ dance, discriminative reading, and valuable time contri¬ buted to the completion of this paper. ii INTRODUCTION The development of the theory of rings has made great advancements during the past fifty years. The topics of interest that concerns us in this paper began with Artin's extension in 1927 of Wedderburn's structure theory of alge¬ bras to rings satisfying the chain conditions. The struc¬ ture theory for algebras over a general field dates from the publication of Wedderburn’s thesis in 1907; the exten¬ sion to rings, from Artin's paper in 1927. The theory of representation was originally concerned with the problem of representing a group by matrices. This was extended to rings and was formulated as a theory of modules by Emmy Noether (1882-1935). The purpose of this paper is to consider the concepts of ideals and radicals in rings with chain conditions and to examine their importance and significance in the develop¬ ment of ring theory in the last fifty years. In that we wish to avoid ambiguity and make this paper largely self- contained, we will define all significant concepts of alge¬ braic structures. As a starting point, it would seem appro¬ priate to define the following principal object of interest in this paper, the notion of a ring. Definition 1.1. A ring R is a nonempty set on which there are two binary operations + and • of addition and multi¬ plication closure on R for which the following conditions are 1 2 satisfied, where it is understood that a,b,c, represent ar¬ bitrary elements of R. (i) a+b=b+a, (commutative law of addition), (ii) (a+b)+c=a+(b+c), (associative law of addition), (iii) there exists an element 0 in R such that a+0=a for every a£R, (existence of a zero), (iv) for each a£R, there exist an element -a*R such that a+(-a)=0, (existence of additive inverse), (v) (a»b)•c=a•(b*c), (associative law of multiplication), (vi) a*(b+c)=a.b+a*c and (b+c)»a=b»a+c.a, (distributive laws). A ring (R,+,*) is said to be a finite or infinite ring if the set R of its elements is a finite or infinite set. By placing restrictions on the multiplication opera¬ tion, several other specialized types of rings are obtained. Definition 1.2. A commutative ring is a ring (R,+,*) in which multiplication is a commutative operation, a*b=b*a for all a,b$R. If a ring R does not satisfy the condition of a commutative ring, it is said to be a non-commutative ring. Definition 1.3. A ring with identity is a ring (R,+,«) in which there exists an identity element for the operation of multiplication normally represented by 1, so that a*1=1* a=a for a£R. Given a ring (R,+,*) with identity 1, an element a£R is said 3 to be invertible or a unit, whenever a possesses an inverse with respect to multiplication. The multiplicative inverse of a is unique, whenever it exists, and will be denoted by a*1, so that a•a~1=a~1•a=l. In the future, the set of all invertible elements of the ring will be designated by the symbol R. Definition 1.4. A nonzero element r of a ring R is called a zero divisor if there exists a nonzero element beR such that rb=0, or a nonzero element aeR such that ar=0. Definition 1.5. A field F is a nontrivial commutative ring with unity with the property that each nonzero element has a multiplicative inverse. Definition 1.6. A commutative ring with unity and with no nonzero divisors of zero is said to be an integral domain. Now we will present some basic properties of rings. Theorem 1.1. If R is a ring, then for any a,b,c€R (i) 0a=a0=0 (ii) a(-b)=(-a)b=-(ab) (iii) (-a) (-b)=ab; and (iv) a(b-c)=ab-ac, (b-c)a=ba-ca Proof (i) Using the definition of zero, 0+0=0, it follows by the distributive laws 0a=(0+0)a=0a+0a=0+0a. Thus, by the cancellation law for the additive group (R,+), we have 0a=0. Similarly, a0=a(0+0)=a0+a0 and by cancellation law for the additive group (R,+), a0=0. 4 (ii) We prove that ab has a(-b) as an additive inverse as follows : a(-b)+ab=a(-b+b)=a0=0. Therefore, a(-b) is an inverse of ab. Hence, -(ab)=a(-b). Similary, -(ab)= -a(b). (iii) Let -(-g)=g for any element g in the additive group (R,+). Then applying this assertion and part (ii) twice, we get (-a)(—b)=—(—a)b=—(—(ab))=ab. (iv) Let a+(-b) be represented as a-b. Then using the distributive laws, a(b+(-c))=ab+a(-c)=ab+(-ac)=ab-ac. Similarly, (b+(-c))a=ba+(-c)a=ba+(-ca)=ba-ca. Q • £ • ]j • To deal with the situation where a subset of a ring constitutes a ring, we make the following definition. Definition 1.7. Let (R,+,#) be a ring. If T is a non¬ empty subset of R such that (T,+,*) is a ring, then T is called a subring of R, and we write T£R. In chapter II we introduce ideals and their operations in the theory of rings. Although it is possible to obtain some interesting conclusions concerning subrings, this con¬ cept, if unrestricted, is too general for most purposes. Thus, we narrow the field and focus attention on a class of subrings with a stronger type of multiplication by an arbitrary ring element. Chapter III will primarily consist of the important radicals in ring theory. There are a number of radicals in existence; several of the more: prominent ones are intro- 5 duced in this chapter. One of these is the Jacobson radical, so named for the brilliant mathematician Nathan Jacobson (1910- ). Chapter IV deals with ideals and radicals in rings with chain conditions. The Noetherian rings are very im¬ portant in this chapter. These satisfy a certain finiteness condition, namely, that every ideal of the ring should be finitely generated. This ring is so named in honor of an outstanding algebraist, Emmy Noether, who first observed the important role that the ascending chain condition plays in the study of rings. From this idea, we are led to con¬ sider a number of results relevant to rings with the descending chain condition. In chapter V, we introduce some important characteri¬ zations of rings with chain conditions. IDEALS IN RINGS In this chapter, the important concepts of ideals that play a central role in ring theory will be discussed. It is appropriate that we start with the following definition. Definition 2.1. A subring N of a ring R is said to be a right ideal if and only if nreN for each neN and reR. Definition 2.2. A subring N of a ring R is said to be a left ideal if and only if rneN for each neN and reR. Definition 2.3. A subring N of a ring R is said to be a two-sided ideal or simply an ideal if and only if for neN and reR imply both rneN, and nreN. This means that the product of any element n of N and any element r of R is an element of N. Before proceeding further, we examine this concept by means of several examples. Example 2.1. Let (R,+,‘) be a ring whose addition and mul¬ tiplication tables are the following: + z a b c z a b c z z a b c z z z z z a a z c b a z a b c b b c z a b z b z b c c b a z c z a b c By examining these tables, we see that {z,a} and {z,c} are subrings of R, but are not ideals, while {z,b} is an ideal of R. In any ring R, the zero subring (the subset contain¬ ing zero only), {Z} where Z=0, and R itself are ideals of R. 6 7 Example 2.2. In this example, we consider the ring of all matrices of order two over the field R of real numbers. Let U be the set of all elements of R£ of the form ^J, a,bfR. Then the product of elements in R2 and U are ele¬ ments of U. Thus, this product shows that U is a right ideal but not a left ideal in R^. Example 2.3. Again consider the ring R of all matrices of order two over the field R of real numbers. Let U be the set of all elements of Rg of the form a,b£R. Thus, the product on the left shows that U is a left ideal and the product on the right indicate U is not a right ideal.
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