Abstract
In this paper, we initiate the study of some annihilator conditions on polynomials which were used by Kaplansky [4] to abstract algebras of bounded linear operators on a Hilbert spaces with Baer condition. On the other hand, p.p. rings were introduced by A. Hattori [3] to study the torsion theory. The purpose of this paper is to introduce near-rings with Baer condition and near-rings with p.p. condition which are somewhat different from the ring case, and to extend a results of Armendarz [1] to polynomial near-rings with Baer condition in somewhat different way of Birkenmeier and Huang [2].