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RADICALS of a RING Approved: Ya'<L/X Major, Professor RADICALS OF A RING approved: yA'<L/X Major, Professor ,/T) flP C Minor Professor 35ir'ector of tne^evarhepartmen. t of MaSHe^ETcc ~~7 Tie an "of The Gra3lia"5e"~ScaooT ± Crawford, Phyllis J., Radicals of a Ring, Master of Science (Mathematics), May, 1971, pp. 48, bibliography, 7* titles. The problem with which this investigation is concerned is that of determining the properties of three radicals de- fined on an arbitrary ring and determining when these radicals coincide. The three radicals discussed are the nil radical, the Jacobson radical, and the Brown-McCoy radical. ,:.n an arbitrary ring R, the prime radical <?( F) ? is delined as the Intersection of all prime Ideals in the ring, Rvery element of £>(R) is nilpotent, that is, If a •; p(f<) then there Is some positive integer n so that aP « o, Thus *>YR) ir= a ri.1J 3deal since every element of 0(R) is nilpotent. If A Ife an Ideal in R such that there is some positive integer K so that a' ~ o for all a e A} then A Is a nilpotent Ideal and A £ ^R)« A. nil radical of R, */>(R) is defined as the sum of all nilpotent ideals in R, Thus '//(R) Is a nil ideal oi. eit contains every nilpotent id.eal of the ring, Then an upper nil radical of R, l{ is defined as the sum of all nil ideals in K so that every nil radical of R is contained in l(. The -tower nil radical of R, is defined as the Intersection ol. 8j.j rij.i i'aaj cais of R so that every nil radical of R con- conf; trucuion of ^(R)., the conclusion reached 1« that <?(R) - £ so that ^ - TF(E) C ??(R) c: IF R is commutative or if R has the descending chain condition for right or left ideals, then st = ^(R) = ??(R) = V. so that there is exactly one nil radical of R. The Jacobson radical of a ring R, j?(R)> is defined as the intersection of the annihilators of the left irreducible modules of R and coincides with the intersection of the annihilators of the right irreducible modules of R. Thus j?(R) coincides with the intersection of all maximal regular left-ideals in R. If P is a left ideal in R such that there is an a e R so that x - xa e P for all x e R, then P is a regular left ideal. Similarly, if x e R and there is some y e R such that x+y+xy=o (x + y + yx = o) then x is right (left) quasi-regular. Thus $(R) consists of those elements of R that are both right and left quasi-regular. Since every nilpotent element is quasi-regular, then K c j?(R) so that £ c ??(R) c K c £(R). if A is an ideal in R and A is considered as a subring of R, then J?(A) = <?(R) If R is a ring with the descending chain condition for right or left ideals, then £ = U = j?(R). In a ring R, an element a e R is G-regular if a e G(a) = {ar - r + 2(xj_ayi - x1y±) [r,xi,yi e R}. The Brown-McCoy radical of R, /ft(R), is defined as the set of all b e R so that the ideal (b) generated by b is G-regular. Thus $(R) c ^(R). if A is an ideal in R and A is considered as a subring of R, then A) = 57?(R) nA. If R is a ring with the descending chain condition for right or left ideals, then s£ = K = #(R) ~ ^(R)- Therefore, in a ring with the descending chain condition for right or left ideals, the nil radical is the only radical of the ring. RADICALS OF A RING- THESIS Presented to the Graduate Council of the North Texas State University in Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE By Phyllis Crawford, B. S, Denton, Texas May, 1971 INTRODUCTION . In an arbitrary ring R, a particular ideal of R can "be defined so that such an ideal is called a radical of R. Since different radicals can be defined on a ring R, the conditions on R under which they coincide will be investi- gated, along with the properties of three of these radicals. Some basic concepts of rings will be used in the following theorems. The sum of two right (left or two-sided) n ideals A and B in a ring R is A + B = {Z) (a. + b. ) | a. e A, b. € B i=l ill- 1 and n ranges over all positive integers}. If {A^} is an in- finite collection of rlght(left or two-sided) ideals in R, T,A. is the ideal whose elements^ consist of finite summations of elements of the A.. The product of two right (left or two- n sided) ideals A.B in R is AB = {S a.b.la. e A, b. e B for 1=1 111 i all positive integers n}. If r e R and A is a right (left or two-sided) ideal in R, then Ar = {ar|a e A} and rA = {ra|a € A}. In general a two-sided ideal in R will be referred to as an Ideal in R. If R is a ring in which the descending chain condition for right (left) Ideals holds, then R Is said to "be right (left) Artinian. If R is a ring in which the ascending chain condition for right(left) Ideals holds, then R is said to be right (left) Noetherian. iii TABLE OF CONTENTS Chapter Page I. NIL RADICALS 1 II. JACOBSON RADICAL 22 III. BROWN-McCOY RADICAL 39 BIBLIOGRAPHY 47 IV CHAPTER I NIL. RADICALS Definition 1.1: An ideal P In a ring R is a prime ideal if for ideals A,B in R, ABjSTP implies A £ P or B«™P. Theorem 1.1: If P is an ideal in a ring R, the follow- ing are equivalent: (I) P is a prime ideal* (ii) If a,b e R such that aRb cP^ then a e P or b e P; (ill) If (a), ("b) are principal ideals in R such that (a)(b)CP, then a € P or b € Pj (iv) If U,V are right ideals in R such that W<=P, then » UcP or V^P; (v) If U,V are left Ideals in R such that W<^P, then U c.P or Vc P. Proof: Assume that P is a prime ideal in a ring R, then let a,b e R such that aRbcp. Then aRb = {arb|r e R}, such that RaRbR £P so that (RaR) (RbR) CP, Then since P is a prime ideal, RaR ^P or RbR«=P. For (a) the principal ideal In R generated by a, (a)^RaR, And for (b) the principal ideal in R generated by b, (b)3c RbR. If RaRc.P, then (a)5cP and since P is prime, (a) CP and thus a e P. If RbR cp, then (b) CP and then (b)cp so that b e P. Then since RaRCP or RbRCp, it follows that a e P or b e P. Therefore, (i) im- plies (ii). 2 Now assume (±i) and let (a), (I)) be principal ideals in r such that (a)(b) £» P, where (a)(b) is the ideal generated by elements of the form a^b^ for e -(a) and b1 € (b). Since ( a),(b) are principal ideals then. aRb (a) (b) so that aRb P and thus a e P or b e P. Therefore, (ii) implies (iii). Assume (iii) and let U,V be right ideals in R such that UV <£ P. Suppose U^ P, then there is some u e U such that u | P. Let v e V. Since UV £ P, then RUV£, P so that UV + RUVi£ P and hence (u)(v) C UV + RUVg^ P. Then, by (iii) for (u)(v)<T p, u e P or v e P. Since u | P, then v e P and since v was arbitrary in V, then V «£ P. Thus (iii) implies (iv). Assume (iii) again and let U,V be left ideals in R such that UVC P. Suppose UJ^iP, then there is some u e U such that u | P, Let v e V, then, similar to the previous•proof u v + ( )( )lSi UV UVR c P so that by (iii), u e P or v e P. Since u P, then v e P and since v was arbitrary in V, then V£ P. Thus (iii) implies (v). Now assume (iv) and let A,B be ideals in R such that ABcp, Since A,B are ideals in R, then A,B are right ideals in R so that by (iv), A P or B P. Thus (iv) implies (i). Similarly, if (v) is assumed and A,B are ideals in R such- that AB ^P, then A,B are left ideals in R. Then by (v), A C p or B <2 P. Thus (v) implies (i). Therefore, all five conditions are equivalent. 3 Definition 1.2; A set M in a ring R is an ro-system if for aflb e M, there is.an x e R such that axb e M. It follows trivially that (j) is an m-system. Theorem 1,2; An ideal P in a ring "R is a prime ideal if and only if P = {x e Rjx | P} is an m-system.
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