Bulletin of the
Korean Mathematical Society BKMS

An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. A ring is strongly nil clean in case each of its elements is strongly nil clean. We investigate, in this article, the strongly nil cleanness of $2\times 2$ matrices over local rings. For commutative local rings, we characterize strongly nil cleanness in terms of solvability of quadratic equations. The strongly nil cleanness of a single triangular matrix is studied as well.

Keywords: $2\times 2$ matrix, strongly nil cleanness, local ring

MSC numbers: 16S50, 16U99