On the Centralizer of an I-Matrix in M 2(R/I), I a Principal Ideal and R a UFD

Abstract

The concept of an I-matrix in the full 2 × 2 matrix ring M 2(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, was introduced in a Marais article that is to be published [10]. Moreover, a concrete description of the centralizer of an I-matrix in M 2(R/I) as the sum of two subrings 𝒮1 and 𝒮2 of M 2(R/I) was also given, where 𝒮1 is the image (under the natural epimorphism from M 2(R) to M 2(R/I)) of the centralizer in M 2(R) of a pre-image of , and where the entries in 𝒮2 are intersections of certain annihilators of elements arising from the entries of . In the present paper, we obtain results for the case when I is a principal ideal⟨k⟩, k ∈ R a nonzero nonunit. Mainly we solve two problems. Firstly we find necessary and sufficient conditions for when 𝒮1 ⊆ 𝒮2, for when 𝒮2 ⊆ 𝒮1 and for when 𝒮1 = 𝒮2. Secondly we provide a formula for the number of elements in the centralizer of for the case when R/⟨k⟩is finite.