I. Made Sulandra, Ichtiarida Mulyasari
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 g(x)= k=1mx2k C(M2(2)) [x], where m is an even integer and is not divided by 3 such that the matrix ring M2( 2) over the field 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 M2( 2) and all roots of g(x) have been found by constructing a new partition of the ring M2( 2), i.e. {{A|A2 = I}, {A|A2 = 0}, {A|A2 = A, I A 0}, {A|A3 = I, I A 0, A2 A}} M2( 2). This idea might be used in the matrix ring M2( p) over the field p integer modulo p, where p is a prime number. © 2020 Author(s).
Universitas Negeri, Malang, Indonesia
