A characterization of minimal prime ideals

Gary F. Birkenmeier, Jin Yong Kim, Jae Keol Park
1998 Glasgow Mathematical Journal  
Let P be a prime ideal of a ring R, O(P) = [a € R\ aRs = 0, for some .? € R\P] and O(P) -{x e R | x" e O(P), for some positive integer n). Several authors have obtained sheaf representations of rings whose stalks are of the form R/O{P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P = O(P). In this paper we derive various conditions which ensure that a prime ideal P -O(P). The property that P = 0{P) is then used to obtain conditions which
more » ... n conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and O(P) are considered. Examples are provided to illustrate and delimit our results. 0. Introduction. Throughout this paper R denotes an associative ring not necessarily with unity, P(R) its prime radical, N(/?) its set of all nilpotent elements and N r (R) its nil radical. R is called a 2-primal ring if P(R) = N(R). We refer to [4], [5], [6], [8], [9], [17], and [19] for more details on 2-primal rings. A proper ideal P of R is called completely prime (completely semiprime) if xy e P (x 2 e P) implies x e P or y e P (x e P). Andrunakievic and Rjabuhin [1] and , independently, Stewart [18] have shown that a reduced ring R (i.e., N(/?) = 0) is a subdirect product of integral domains. Thus a proper ideal is completely semiprime if and only if it is an intersection of completely prime ideals. All prime ideals are taken to be proper ideals. Let A' be a nonempty subset of R, then R , 1{X) and r(X) denote the ideal of R generated by X, the left annihilator of X in R, and the right annihilator of X in R, respectively. Let P be a prime ideal. The following definitions are fundamental to the remainder of our discussion: O P = {a e R | as = 0, for some s e R\P] = Op = {x e R | x" e Op, for some positive integer«), N P = {y e R \ ys e P(R), for some s e R\P], O(P) = {aeR\aRs = 0, for some s e R\P] = {J s€R . p t( Rs )> O(P) = {x e R | x" e O(P), for some positive integer n], and N(P) = {yeR\yRsC. P(R), for some . ? e R\P). Observe that 0{P) c N(P) c p. Furthermore if P is a completely prime ideal, then Op^P and N P c p. EXAMPLE 0.1. Let F b e a field and Glasgow Math. J. 40 (1998) 223-236. https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s0017089500032547 fatcat:zstutaljfjfu5ff6k3k3ygveyu