The Centralizer of an I-Matrix in M2(R/I), R a UFD
Abstract
The concept of an I-matrix in the full 2 × 2 matrix ring M2(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix 
in M2(R/I) as the sum of two subrings 𝒮1 and 𝒮2 of M2(R/I), where 𝒮1 is the image (under the natural epimorphism from M2(R) to M2(R/I)) of the centralizer in M2(R) of a pre-image of 
, and the entries in 𝒮2 are intersections of certain annihilators of elements arising from the entries of 
. It turns out that if R is a PID, then every matrix in M2(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.
This research is part of the author's research for her doctoral dissertation which was conducted at Stellenbosch University under the direction of L. van Wyk. The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the National Research Foundation.
