A class of formal matrix rings

Nil-clean and strongly nil-clean rings

“…Following [30], for n ≥ 2 and for s ∈ C(R), the n × n formal matrix ring over R defined by s, denoted as M n (R; s), is the set of all n × n matrices over R with usual addition of matrices and with multiplication defined below: for (a ij ) and (b ij ) in M n (R; s),…”

Nil-clean and unit-regular elements in certain subrings of ${\mathbb M}_2(\mathbb Z)$

“…sy b , and that Z sZ sZ Z ∼ = M 2 (Z; s), the formal matrix ring defined in [9] (see [9], Proposition 4 (3)). Hence, by [9], Example 23,…”

Strong $J$-Cleanness of Formal Matrix Rings

“…The question of when a matrix ring M n (R) is strongly clean has been discussed by several authors (see [2,1,3,4,9,11]). Recently, Tang and Zhou [9,10] completely characterized the local rings R for which the formal matrix ring M 2 (R; s) is strongly clean. In this article, we are motivated to investigate the strong J-cleanness of the formal matrix rings over a local ring.…”

“…This ring is denoted by M 2 (R; s) in [10]. [10]). Some properties of M 2 (R; s) were studied in [10].…”

“…In what follows, the ring of integers modulo n is denoted by Z n , and we write M n (R) (resp. T n (R)) for the rings of all (resp., all upper triangular) n × n matrices over the ring R. For s ∈ C(R), s : [10,Lemma 10]). For elements a, b in a ring R, we use the notation a ∼ b to mean that a is similar to b, that is, b = u −1 au for some u ∈ U (R).…”