A characterization of minimal prime ideals

Abstract

Abstract. 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 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. 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 <X> 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:

n-Normal residuated lattices

“…L. Leuştean (Leuştean, 2005) introduced the notion of O-filters in BL-algebras as the dual of o-ideals studied by Cornish. O-ideals are the lattice version of the following ideals in rings: if R is a ring, then O(P ) = {a ∈ R| as = 0, for some s ∈ R \ P }, where P is a prime ideal of R. O-ideals are used for obtaining sheaf representations of different classes of rings (Birkenmeier, Kim, and Park, 1998, 2000, Hofmann, 1972. Let A be a residuated lattice and F be a filter of A.…”