A Note on Strongly Clean Matrix Rings

Strongly P-Clean Rings and Matrices

“…Recently, strong cleanness has been extensively studied in the literature (cf. [1][2][3][4][5], [8], [10], [12], [13]). As is well known by [9] that, every 2×2 matrix A over a field satisfies the conditions: A = E + W, E is similar to a diagonal matrix, W ∈ M 2 (R) is nilpotent and E and W commute.…”

On Strongly Nil Clean Matrices

“…There are detailed discussions on these questions in the literature (cf. [1][2][3][4], [8] and [10][11]). Let a ∈ R. l a : R → R and r a : R → R denote, respectively, the abelian group endomorphisms given by l a (r) = ar and r a (r) = ra for all r ∈ R.…”

“…11 , a 22 , a 33 , a 13 ∈ R}.Then T 3 (R) forms a 3 × 3 subring of the ring of all 3 × 3 lower triangular matrices over R under the usual additions and multiplications. Now we characterize the strongly cleanness of such rings, which also extend[5, Theorem 3.1].…”

STRONGLY NIL CLEAN MATRICES OVER R[x]/(x2-1)

“…Strongly clean matrices over commutative local rings were extensively studied by many authors from very different view points (cf. [1][2], [4][5][6][7] and [9][10][11][12]). In [5], Diesl introduced the concept of strongly nil cleanness.…”

“…Also we note that one studied full strongly clean matrices only for the strongly clean matrix rings (cf. [7,Theorem 3.3 and Theorem 3.4]). We will consider the strongly nil cleanness for a single matrix even the matrix rings not having such properties.…”