ABSTRACT
Suppose R is a commutative ring with identity. This paper deals with the problem of determining conditions under which R can be embedded in a zero‐dimensional ring. Work on this question was begun by Arapovic in [1]-[3] and continued by Gilmer and Heinzer in [7],[8]. Essentially one set of conditions equivalent to embeddability is known. This set consists of two conditions (which we label as (A1) and (A2)) and is due to Arapovic; they are stated in Theorem 3.1. In Theorem 4.1 we establish a new criterion for embeddability of R. It states that R is embeddable in a zero‐dimensional ring only if, for each x in R, there is a positive integer n (which may depend upon x) such that xn and x n + 1 have the same annihilator in R. We consider applications of this criterion both for known results concerning embeddability and for some open questions. The question of whether the named criterion is sufficient for embeddability in a zero‐dimensional ring has not been resolved.
Dedicated to Jim Huckaba on the occasion of his retirement
