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IDEALS and RADICALS Ll|||! IDEALS AND RADICALS ll|||! lyiiiiiiiii Ul!ll«l B. DE LA F BIBLIOTHEEK TU Delft P I960 5183 1 656232 IDEALS AND RADICALS 1 I * \ i IDEALS AND RADICALS PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 25 NOVEMBER 1970 TE 14.00 LTUR DOOR BENJAMIN DE LA ROSA MASTER OF SCIENCE GEBOREN TE SMITHFIELD /^éo ^"/^^ N. V. DRUKKERIJ J. J. GROEN EN ZOON - LEIDEN DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. F. LOONSTRA. * Aan Elsje Hiermee betuig ek my dank aan die SUID-AFRIKAANSE WETENSKAPLIKE EN NYWERHEIDNAVORSINGSRAAD vir die toekenning van 'n beurs tydens my studie in Delft CONTENTS CONVENTIONS 1 CHAPTER I INTRODUCTION 3 CHAPTER II PRIME IDEALS AND RADICALS 8 2.1. The radicals which contain the Baer-McCoy radical . 8 2.2. A new radical class which contains p 9 CHAPTER III QUASI-SEMI-PRIME IDEALS AND THE QUASI-RADICAL OF AN IDEAL 13 3.1. Quasi-semi-prime ideals 13 3.2. The quasi-radical of an ideal 17 CHAPTER IV QUASI-RADICAL RINGS AND THE BAER-MCCOY RADICAL CLASS 19 4.1. The quasi-radical of a ring 19 4.2. Quasi-radical rings and the Baer-McCoy radical class. 20 CHAPTER V RINGS IN WHICH ALL IDEALS ARE QUASI-SEMI-PRIME 25 5.1. The A-radical of a ring 25 5.2. Quasi-semi-prime ideals and rings of matrices 31 DUTCH SUMMARY 36 CONVENTIONS By a ring we mean an associative ring, which, unless the contrary is stated, does not necessarily possess a unity and is not necessarily commu­ tative. A two-sided ideal in a ring is referred to simply as an ideal. Rings and ideals are denoted by capital letters. A ring R with more than one element is called a simple ring if its only ideals are the trivial ideals (0) andR. The union of a set {Ai\ iel} of ideals in a ring is the set of all sums of elements from different Ai, each sum containing only a finite number of non-zero terms. The product of two ideals A and 5 in a ring is the set AB={Y,aibi\aieA,bieB}, it being understood that all finite sums of one or more terms are to be included. Products of more than two factors are defined inductively. A ring R is said to be nilpotent if there exists a positive integer n such that R" = {Q)). If R^ = {0), then R is called a trivial ring. An element x of a ring is called a nilpotent element if there exists a positive integer n such that jc" = 0. A nil ring is a ring in which all elements are nilpotent. CHAPTER I INTRODUCTION The origins of ideal theory were closely linked up with the number theoretical notion of division, and accordingly the theory could not escape the influence of the fundamental concept of prime number. The desired counterpart was introduced by EMMY NOETHER in [15], where she defined the concept of a prime ideal in a commutative ring. Her two equivalent definitions may be formulated as follows: DEFINITION 1.1. An ideal P in a commutative ring R is called a prime ideal if from abeP, where a, beR, it follows that aeP or beP. DEFINITION 1.2. An ideal P in a commutative ring R is said to be prime, if the following condition is satisfied: If A and B are ideals in R such that AB^P, then Ac p or Be p. The latter was taken over by KRULL [8 ] for the case of an arbitrary ring, and this adoption has come to play an important role in the de­ velopment of ideal theory. Two further aspects which are important for our purposes are con­ tained in KRULL [7]. Firstly, he observed that an ideal ƒ" in a commutative ring R is prime, if and only if the complement C(P) of P in /? is a multi­ plicative system, that is, aeC{P) and beC{P) imply that abeC{P). Secondly, he introduced the concept of a semi-prime ideal in a commu­ tative ring: An ideal 5 in a commutative ring is called a semi-prime ideal if a"eS for some positive integer n, implies that asS. This definition, in which we may obviously take n = 2, leads in a natural way to the following generalization to the case of an arbitrary ring (cf. [13]). DEFINITION 1.3. An ideal S in a ring R is called a semi-prime ideal if from A^^S, where A is an ideal of R, it follows that A^S. MCCOY [11 ] proved that an ideal P of a ring R is prime if and only if the following condition is satisfied: If a and b are elements of R such that aRb^P, then aeP or beP. This fact inspired his generalization of 3 the concept of a multiplicative system to that of an m-system which he defined as a system M of elements of R with the property that aeM and beM imply that axbeM for some xeR. The importance of this concept lies in the fact that an ideal P in a ring R is prime if and only if the complement of P in i? is an m-system; an exact parallel of the role of multiplicative systems with respect to prime ideals in commutative rings. A further generalization in this respect was suggested by BROWN and MCCOY in [2] and carried out by the latter in [13] where he introduced the concept of an n-system to fill the same complementary role with respect to semi-prime ideals: An n-system N of a ring /? is a subset of R with the property that aeN implies the existence of an element x in R such that axaeN. McCoy proved that for every aeN there exists an 7M-system M in i? such that aeM and M^N. This relationship between wi-systems and «-systems represents a crucial point in his proof of the following important result on the relationship between semi-prime ideals and prime ideals. LEMMA 1.4. An ideal S in a ring R is a semi-prime ideal if and only if S can be represented as an intersection of prime ideals of R. In [11] McCoy defined the prime radical of an ideal A in a. ring R as the set of all elements r of R with the property that every w-system of R which contains r meets A. The set so defined for the zero ideal is called the prime radical of the ring R. He proved that this radical coincides with the intersection of all prime ideals in R. The prime radical coincides with the Baer lower radical introduced in [1]; a fact which was proved in­ dependently by LEVITZKI [10] and NAGATA [14]. We shall refer to this radical as the Baer-McCoy radical and denote it by ^{R). The theory of radicals has come a structure revealing way and since the coordinating definition by KUROSH in the early 1950's, this theory has matured into a confluence of elegance and generality. The definition referred to is included here for reference (cf. [4]). Let (7 be a certain property that a ring may possess. A ring R is called a a-ring if it has the property a. An ideal ^4 of a given ring is called a a-ideal if A, viewed as a ring, is a a-ring. A ring which does not contain any non-zero c-ideals is said to be a-semi-simple. DEFINITION 1.5. A property a is called a radical property if the following three conditions are satisfied: (1) A homomorphic image of a a-ring is a a-ring. 4 (2) Every ring R contains a a-ideal S which contains every other a-ideal of R. (3) The factor ring R/S is a-semi-simple. The unique maximal «r-ideal S of i? is called the a-radical of R and is denoted by a(R). A a-ring is its own a-radical. Such a ring shall be termed a a-radical ring. The requirement (2) ensures that (0) is a a- radical ring with respect to any given radical property a. Obviously a ring is a-semi-simple if and only if a(R) = (0). A given radical property and the class of all rings which are radical with respect to it, will be denoted by the same symbol. Radical properties are compared according to inclusion: If p and a are two radical properties, we shall write p < a to indicate the fact that the class of all p-radical rings is contained in the class of all a-radical rings. It can be verified that this condition is equivalent to either of the following two. (i) p{R)^a(R) for every ring R. (ii) Every a-semi-simple ring is p-semi-simple. Two important types of radical properties have been constructed and widely used, the so-called lower and upper radical properties. The two constructions are briefly sketched below (cf. [4, 18]). Let .5f be a non-empty homomorphically closed class of rings. Define JSfi=if. Assume that iP^, has been defined for every ordinal number a such that Ka<)?, and define if^ to be the class of all rings R such that every non-zero homomorphic image of R contains a non-zero ideal which is a ring of if^, for some a.<fi.
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