Sexauer, Norman E.||Beach, James W.
M.S. (Master of Science)
Department of Mathematics
On October 28, 1944, Nathan Jacobson presented to the American Mathematical Society his paper entitled "The Radical and Semi-simplicity for Arbitrary Rings." He defined in the first few pages of his paper a concept which has become known in the literature as the Jacobson radical. This concept is the focal point of this study. The radical of an arbitrary ring is important to ring theory as a basis upon which to build a structure theory for arbitrary rings. Prior to the Jacobson radical there were several characterizations of the concept which proved Inadequate for the development of a satisfactory structure theory. Two of these are as follows: 1) The radical of a ring which satisfies the descending chain condition is the join of the nil ideals of A. 2) The radical of a ring A with an identity is the Intersection of the maximal right (left) ideals of A. Let z be an arbitrary element of a ring A. Then if there exists an element z’ in A such that z+z'zz''O then z is called right quasi-regular and z' is its right quasi-inverse. A right ideal is quasi-regular if all of its elements are right quasi-regular. The Jacobson radical of a ring Is defined to be the join of the quasi-regular right ideals of the ring. Moreover, the radical is a two-sided quasi-regular ideal of the ring. Left quasi- regularity, quasi-regular left ideal and left radical could be defined in a similar manner. An Important result to be noted is that the left radical and the right radical coincide. As one might expect under the proper conditions the Jacobson radical coincides with the above characterizations of the radical. The radical of an algebra is defined precisely the same as for an arbitrary ring. The radical of a ring of n by n matrices over a ring A is the set of n by n matrices over R the radical of A. A ring whose radical consists of zero only is called a semi-simple ring. The ring of integers is such a ring. An important example of a semi-simple ring is the ring A/R where R is the radical of the ring A. The concept of semi-simplicity quickly leads to the study of primitive rings, irreducible rings of endomorphisms and dense rings of linear transformations over a vector space. All of these concepts have undergone considerable study in recent years.
Bednar, Louis Edward, "A study of the Jacobson radical" (1965). Graduate Research Theses & Dissertations. 744.
viii, 58 pages
Northern Illinois University
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