When is a 2 × 2 Matrix Ring Over a Commutative Local Ring Quasipolar?

Abstract

A ring R is quasipolar if for any a ∈ R, there exists p 2 = p ∈ R such that p ∈ comm2(a), p + a ∈ U(R) and ap ∈ R qnil . In this article, we determine when a 2 × 2 matrix over a commutative local ring is quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be quasipolar. Consequently, we obtain several equivalent conditions for the 2 × 2 matrix ring over a commutative local ring to be quasipolar. Furthermore, it is shown that the 2 × 2 matrix ring over the ring of p-adic integers is quasipolar.