On Polynomials Over Rickart Rings and Their Generalizations I

This paper continues the investigation of polynomials and formal power series over a ring with various annihilator conditions which were originally used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. Results of Armendariz on polynomial rings over a PP ring are extended to analogous annihilator conditions in nearrings of polynomials and nearrings of formal power series. These results are somewhat striking since, in contrast to the polynomial ring case, the nearring of polynomials or formal power series has substitution for its “multiplication” operation. These investigations provide an alternative viewpoint in illustrating the structure of polynomials and formal power series. Extensions of Rickart rings to formal power series rings are also discussed. The author was partially supported by the National Science Council, Taiwan under the grant number NSC 93-2115-M-143-001.