Study on Ring: g(x)-nil-clean and Strongly g(x)-clean, but not Strongly g(x)-nil-clean

摘要

Let g(x) ∈ C(R)[x] be a polynomial in variable x with coefficient in the centre of a ring R with unity. Every strongly g (x) -nil-clean-ring is g (x) -nil-clean and also strongly g (x)-clean. But, its converse is incorrect. We construct a polynomial {formula}, where m is an even integer and is not divided by 3 such that the matrix ring M_2(Z_2) over the field Z_2 integer modulo 2 is g(x)-nil-clean and strongly g(x)-clean, but not strongly g(x)-nil-clean. All nilpotent elements of M_2(Z_2) and all roots of g(x) have been found by constructing a new partition of the ring M_2(Z_2), i.e. {formula}. This idea might be used in the matrix ring M_2(Z_p) over the field Zp integer modulo p, where p is a prime number.