Intersection Irreducible Ideals of a Non-Commutative Principal Ideal Domain

Abstract

Let R always denote a fixed non-commutative principal ideal domain. A right (left) ideal aR (Ra) is termed right (left) ∩ irreducible provided it is not the intersection of two right (left) ideals that properly include it. In this case, the element a is called right (left) ∩ irreducible.Since R satisfies the A.C.C. for right ideals every right ideal aR can be written in the form aR = a,1R ∩ a2R ∩ … ∩ anR, where atR properly include aR and is right ∩ irreducible, i = 1,2, … ,n. We shall investigate properties (including primary properties) of right ∩ irreducible one-sided and two-sided ideals of R. These properties will depend on the results given in (1) and (2, chapter III).An element a is irreducible if it is not zero or a unit and has no proper factors. In this case aR (Ra) is a maximal right (left) ideal.