The centralizer of an I-matrix in M₂(R/I), R a UFD

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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 B 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 B, and where the entries in S₂ are intersections of certain annihilators of elements arising from the entries of B. 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.