On two open problems about strongly clean rings
On two open problems about strongly clean rings
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DOI：
10.1017/s0004972700034493
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发表时间：
2004-10
影响因子：
0.7
通讯作者：
Zhou Wang;Jianlong Chen
中科院分区：
文献类型：
--
作者：
Zhou Wang;Jianlong Chen
一个环称为强干净环，如果每个元素都是幂等元与可交换单位之和。1999年，Nicholson提出了一个问题：是否每一个半完全环都是强干净的，以及强干净环的矩阵环是否是强干净的。本文证明了：若R = {m/n ∈ R：n是奇数}，则M2（R）是半完全环但不是强clean环.因此，我们对这两个问题都给出了否定的答案。证明了环R上的每个上三角矩阵环都是强clean的。
A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n ∈ ℚ: n is odd}, then M2(R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.