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/-matrix B i...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/-matrix B in M_2（R/I） as the sum of two subrings ,S_1 and ,S_2 of M_2（R/I）, where S_1 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 B, and the entries in S_2 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 M_2（R/I） is an/-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.展开更多
文摘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/-matrix B in M_2（R/I） as the sum of two subrings ,S_1 and ,S_2 of M_2（R/I）, where S_1 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 B, and the entries in S_2 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 M_2（R/I） is an/-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.
