QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS
QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS
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DOI：
10.4134/bkms.2014.51.3.813
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发表时间：
2014-05
影响因子：
0.5
通讯作者：
Jian Cui;Xiaobin Yin
中科院分区：
文献类型：
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作者：
Jian Cui;Xiaobin Yin
抽象的。一个环R称为拟极环，如果对每个a∈R，存在sp2=p∈R使得p∈comm 2R(A)，a+p∈U(R)和ap∈R qnil.拟极环类恰好位于强π-正则环类和强清洁环类之间.在这篇文章中，我们确定了局部环上的2×2矩阵何时是拟极的。给出了2×2矩阵环为拟极环的充要条件。1.引言在本文中，环R与单位元结合，模M是酉模。设M_n(R)是环R上的n×n矩阵环，In是M_n(R)上的n×n单位根，设M_n(R)是R上的n×n矩阵环，In_n是M_n(R)的n×n单位矩阵，设M_n(R)是R上的n×n矩阵环，∈_a和r_a分别表示R的左、右乘法给出的交换群同态.我们对模M的自同态环记为π(M)，称环R是强∈-正则的，如果对每个a⊇R，链Ar⊇a 2 R⊇··终止(或等价地，链Ra Ra
Abstract. A ring R is called quasipolar if for every a ∈ R there existsp 2 = p ∈ R such that p ∈ comm 2R (a), a + p ∈ U(R) and ap ∈ R qnil .The class of quasipolar rings lies properly between the class of stronglyπ-regular rings and the class of strongly clean rings. In this paper, wedetermine when a 2 ×2 matrix over a local ring is quasipolar. Necessaryand sufficient conditions for a 2 × 2 matrix ring to be quasipolar areobtained. 1. IntroductionThroughout the paper, rings R are associative with unity and modules Mare unitary modules. For an element a ∈ R, l a and r a denote the abelian groupendomorphisms of R given by left and right multiplication by a, respectively.The symbols U(R) and J(R) stand for the group of units and the Jacobsonradical of R. Let M n (R) be the n×n matrix ring over R and I n be the n ×nidentity matrix of M n (R). We write end(M) for the endomorphism ring of amodule M.Recall that a ring R is called strongly π-regular if for every a ∈ R, thechain aR ⊇ a 2 R ⊇ ··· terminates (or equivalently, the chain Ra ⊇ Ra