When is every matrix over a division ring a sum of an idempotent and a nilpotent
When is every matrix over a division ring a sum of an idempotent and a nilpotent
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DOI：
10.1016/j.laa.2014.02.047
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发表时间：
2014-06
影响因子：
1.1
通讯作者：
M. Koşan;Tsiu-Kwen Lee;Yiqiang Zhou
中科院分区：
文献类型：
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作者：
M. Koşan;Tsiu-Kwen Lee;Yiqiang Zhou
如果一个环的每一个元素都是幂等元素和幂零元素的和，那么这个环就被称为零干净环。针对S. brez等人在[1]中的问题，证明了除环D上的n× n矩阵环当且仅当D = f2是一个零干净环。作为结果，证明了强正则环R上的nxn矩阵环当且仅当R是布尔环是零干净环，半局部环R当且仅当其Jacobson根J (R)为零且R/J (R)是矩阵环在F 2上的直积是零干净环。
A ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1], we prove that the n× n matrix ring over a division ring D is a nil-clean ring if and only if D≅ F 2. As consequences, it is shown that the n× n matrix ring over a strongly regular ring R is a nil-clean ring if and only if R is a Boolean ring, and that a semilocal ring R is nil-clean if and only if its Jacobson radical J (R) is nil and R/J (R) is a direct product of matrix rings over F 2.