When is the 2×2 matrix ring over a commutative local ring strongly clean?
When is the 2×2 matrix ring over a commutative local ring strongly clean?
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DOI：
10.1016/j.jalgebra.2005.08.005
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发表时间：
2006-07
影响因子：
0.9
通讯作者：
Jianlong Chen;Xiande Yang;Yiqiang Zhou
中科院分区：
文献类型：
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作者：
Jianlong Chen;Xiande Yang;Yiqiang Zhou
如果 R 的每个元素都是幂等单元和可交换单元的和，则具有恒等性的环 R 称为强清洁环。局部环非常干净。矩阵环何时达到高度清洁是未知的。然而从[J. Chen, X. Yang, Y. Zhou，关于强干净矩阵和三角矩阵环，预印本，2005] 对于任何素数 p，2×2 矩阵环 M2(Zˆp) 是强干净的，其中 Zˆpi 是 p 进整数的环，但 M2(Z(p)) 不是强干净的，其中 Z(p) 是 Z 在 p 生成的素理想上的定位。令 R 为交换局部环。获得了 R 中简单二次方程的可解性标准，以使 M2(R) 具有强干净性。因此，M2(R) 是强干净的，当且仅当 M2(R[x]) 是强干净的，当且仅当 M2(R[x]/(xn)) 是强干净的，当且仅当 M2(RC2) 是强干净的。
A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local rings are strongly clean. It is unknown when a matrix ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005] that for any prime number p, the 2×2 matrix ring M2(Zˆp) is strongly clean where Zˆpis the ring of p-adic integers, but M2(Z(p)) is not strongly clean where Z(p)is the localization of Z at the prime ideal generated by p. Let R be a commutative local ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for M2(R) to be strongly clean. As consequences, M2(R) is strongly clean iff M2(R[x]) is strongly clean iff M2(R[x]/(xn)) is strongly clean iff M2(RC2) is strongly clean.