标题： 环上每个矩阵都是两个幂等元和一个幂零元的和 显示英文标题

标题： Rings over which every matrix is the sum of two idempotents and a nilpotent

摘要： 一个环$R$是（强）2-幂等-幂零环，如果$R$中的每个元素都是两个幂等元和一个幂零元（可交换）的和。讨论了这类环的基本性质。设$R$是一个 2-原始环。如果$R$是强 2-幂等-幂零环，我们证明对于所有$n\in {\Bbb N}$，$M_n(R)$是 2-幂等-幂零环。我们还证明，对于有界指数的强 2-幂等-幂零环，矩阵环是 2-幂等-幂零环。这些结果提供了许多环类，其中每个矩阵都是两个幂等元和一个幂零元的和。 显示英文摘要

摘要： A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we show that $M_n(R)$ is 2-nil-clean for all $n\in {\Bbb N}$. We also prove that the matrix ring is 2-nil-clean for a strongly 2-nil-clean ring of bounded index. These provide many classes of rings over which every matrix is the sum of two idempotents and a nilpotent.