Reflective 𝑁-prime rings with the ascending chain condition

Abstract

Introduction. In an important paper [3] A. W. Goldie proved that a ring is a prime ring with certain chain conditions if and only if it has a quotient ring which is a total matrix ring Dn, where D is a division ring. For a ring R let N(R) denote the nil radical of R. We say that a ring K is completely primary if K has an identity and K/N(K) is a division ring. In ?1 we characterize those rings R which satisfy the ascending chain condition (A.C.C.) for right and left ideals and which have a quotient ring Q(R) of the form Kn where K is completely primary and where Q(R/N(R)) -Q(R)/N(Q(R)). These rings are defined in ?1 as reflective N-prime rings. In ?2 it is proved that if R is reflective N-prime then R[x], where x is a commutative indeterminate, is reflective N-prime. 1. Reflective N-prime rings. Throughout this paper, R will denote a ring which satisfies the A.C.C. for right and left ideals(2). This, of course, implies that chain conditions (1) and (2) of [3] are satisfied in R and R/N(R). If S is a subset of R then S will denote the image of S under the natural homomorphism from R to R/N(R). In particular, R=R/N(R). DEFINITION 1.1. A ring R is termed N-prime if R/N(R) is a prime ring. DEFINITION 1. 2. Let pi and P2 be ideals of a ring R. Then R is called strongly N-prime if PlP2 =0, P' 0 implies p2CN(R) and P2$0 implies p1CN(R). DEFINITION 1.3. Let q be an ideal of a ring A. Then A is said to be qreflective if the following condition is satisfied: an element a is regular in A if and only if a+q is regular in A/q. If A is N(A)-reflective, then A is termed reflective. We shall say that A is reflective N-prime if A is both reflective and Nprime. STATEMENT 1.1. If R is strongly N-prime then R is N-prime. Proof. Suppose P1P2CN(R) for ideals pi and P2 of R. Then (P1p2) n=0 for some positive integer n. From this product let pi,, p2i, . * *, pi, be the smallest subcollection whose product pipi ... pi, is zero. Since pi, Pik,is not zero then i>.CN(R). Thus either b, or -b is in N(R) and R is Drime.