Quotient rings of semiprime rings with bounded index

Abstract

We say that a ring R has bounded index if there is a positive integer n such that a" = 0 for each nilpotent element a of R. If n is the least such integer we say R has index n. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ring Q of a semiprime ring R with bounded index. Several special cases of this situation have already received attention in the literature. If R satisfies a polynomial identity [1], or if every nonzero right ideal of R contains a nonzero idempotent [18] then it is known that Q is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index of R. On the other hand if R is reduced (that is, has index 1) then Q is a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).THEOREM. Let R be a semiprime ring of index n and let Q be its maximal right quotient ring. Then Q = Q i ® Q 3 where Q 3 is a biregular right self-injective ring of type III, and where Q t is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being at most n x n.We also show that the natural attempt to generalize this theorem to right nonsingular rings with bounded index breaks down quite badly. We give an example where Q is itself type III but not biregular, and show (Proposition 15) that every regular right self-injective ring of type I is the maximal quotient ring of a ring with index at most 2. The only part of the theorem which remains intact is that Q still cannot have a direct factor of type II (Theorem 17).None of the techniques used in the special cases mentioned above seems to work for general semiprime rings with bounded index, so we must start from scratch. In section 1 we develop some basic properties of rings with bounded index (most of these are probably well-known but I have been unable to find a suitable reference for them). For our purposes the main point to emerge is that if R is semiprime with bounded index then R is (right) nonsingular. This implies that the maximal right quotient ring Q of R is regular and right self-injective, and means that we have a decomposition of Q into a direct product of type I, II and III rings (as described in [6] and [7]). In section 2 we show how properties of the maximal quotient ring can induce nilpotent elements in the original ring (Lemma 7). We use this to simplify the decomposition of Q, and thus obtain the results mentioned earlier.In what follows all rings are associative but they need not have an identity element. A Glasgow Math. J. 23 (1982) 53-64. 54JOHN HANNAH ring is semiprime if it has no nonzero nilpotent ideals. We denote the right annihilator of a subset X of a ring R by r R (X) or si...

Uniquely Morphic Rings

“…The argument is similar to the proof of [2,Theorem 7.2]. First notice that R is right nonsingular by [3,Proposition 4]. Suppose that g is not central.…”

Hulls of semiprime rings with applications to C∗-algebras

“…Hence there is I i P R with R (I i ) = 0 and e i I i ⊆ R. Therefore a k i e i I i = 0 and e i I i ⊆ r R (a k i ). Since R has bounded index at most n, r R (a k i ) = r R (a n i ) by [39,Proposition 2], so e i I i ⊆ r R (a n i ). Thus a n i e i I i = 0, hence a n i e i = 0 for each i.…”