Abstract
An element of a ring $R$ is called nil-clean if it is the sum of an idempotent and a nilpotent element. A ring is called nil-clean if each of its elements is nil-clean. S. Breaz et al. in [1] proved their main result that the matrix ring $\mathbb{M}_{ n}(F)$ over a field $F$ is nil-clean if and only if $F\cong \mathbb{F}_2$, where $\mathbb{F}_2$ is the field of two elements. M. T. Ko\c{s}an et al. generalized this result to a division ring. In this paper, we show that the $n\times n$ matrix ring over a principal ideal domain $R$ is a nil-clean ring if and only if $R$ is isomorphic to $\mathbb{F}_2$. Also, we show that the same result is true for the $2\times 2$ matrix ring over an integral domain $R$. As a consequence, we show that for a commutative ring $R$, if $\mathbb{M}_{ 2}(R)$ is a nil-clean ring, then dim$R=0$ and char${R}/{J(R)}=2$.
Highlights
Covering arrays have been explored as a method to reveal the presence of faults caused by interactions among components in a complex system [4, 8], they are inadequate to determine which interaction(s) account for the faulty behaviour
While well studied, covering arrays do not enable one to determine the specific levels of factors causing the faults; locating arrays ensure that the results from test suite execution suffice to determine the precise levels and factors causing faults, when the number of such causes is small
Each operates by repeating subarrays; because we want different interactions to appear in different sets of rows, such recursions for locating arrays necessitates more ingredients than for covering arrays
Summary
Covering arrays have been explored as a method to reveal the presence of faults caused by interactions among components in a complex system [4, 8], they are inadequate to determine which interaction(s) account for the faulty behaviour. To formulate arrays for testing, we limit both the number of interactions causing faults and their strengths. At present unless the number of factors is small, the observation in [7] that covering arrays of higher strength provide examples of locating arrays serves as the main device for their construction. Each operates by repeating subarrays; because we want different interactions to appear in different sets of rows, such recursions for locating arrays necessitates more ingredients than for covering arrays. We focus on constructions for (1, 2)-locating arrays in which all factors have the same number v of levels. (the difference list) contains every element of Γ λ times
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