Elements of Minimal Prime Ideals in General Rings

Abstract

Let R be any ring; a ∈ R is called a weak zero-divisor if there are r, s ∈ R with ras = 0 and rs = 0. It is shown that, in any ring R, the elements of a minimal prime ideal are weak zero-divisors. Examples show that a minimal prime ideal may have elements which are neither left nor right zero-divisors. However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors. The union of the minimal prime ideals is studied in 2-primal rings and the union of the minimal strongly prime ideals (in the sense of Rowen) in NI-rings.