On ring extensions for completely primary noncommutative rings

Abstract

0. Introduction. It is the authors' purpose in this paper to initiate the study of ring extensions for completely N primary noncommutative rings which satisfy the ascending chain condition for right ideals (A.C.C.). We begin here by showing that every completely N primary ring R with A.C.C. is properly contained in just such a ring. This is accomplished by first showing that R[x], x an indeterminate where ax = xa for all a E R, is N primary and then constructing the right quotient ring Q(R[x]). The details of these results appear in ?? 1, 7 and 8. The corresponding results for the commutative case are given by E. Snapper in [7] and [8]. If R c A, where A is completely N primary with A.C.C. then,from the discussion in the preceding paragraph, it would seem natural to examine the structure of R(or) when a eA and ac = ca for all a eR in the cases where a is algebraic or transcendental over R. These structures are determined in ?? 6 and 8 of the present paper. The definitions and notations given in [2] will be used throughout this paper. As in [2], for a ring R, N or N(R) denotes the union of nilpotent ideals(') of R, P or P(R) denotes the set of nilpotent elements of R and J or J(R) the Jacobson radical of R. The letter H is used for the natural homomorphism from R to R/N = R. If B is a subset of R then B denotes the image of B under H. If N = P in R and if R' is a ring contained in R then N(R') = NrlR' and k' = R'/N(R'). Thus we consider the contraction of H on R' as the natural homomorphism from R' onto R'/N(R'). Unlike the commutative case, the results of this paper will at times depend on the three conditions (i), (ii) and (iii) of [2, ? 3]. Therefore, we make the following definition. DEFINITION 0.1. A ring R with identity is called an extendable ring if it satisfies the three conditions: (i) P(q) is an ideal when q is a right P primary ideal(2). (ii) P(R) = N(R). (iii) The nontrivial completely prime ideals of R/N are maximal right ideals.