Baer and quasi-Baer annihilator conditions for nearrings and rings

A ring with unity is called Baer (quasi-Baer) if the left annihilator of each nonempty set (ideal) is generated by an idempotent element. The origins of the class of Baer rings evolved as an abstraction of the strictly algebraic properties of von Neumann algebras. This concept has been extended to nearrings. However in the classes of nearrings and rings without unity, the Baer concept splits into at least four distinct classes and at least eight classes for the quasi-Baer concept (see below). We investigate certain nearring and ring decompositions induced by Baer or quasi-Baer annihilator conditions. Examples are provided to illustrate and delimit our results.