交换局部环上的强二和3×3矩阵

摘要

称环R的元有强二和性质,如果它可以写成环中两个可交换单位的和.如果环R的每个元都有强二和性质,则称环R为强二和环.本文研究了3×3阶矩阵环的两个子环C(R)和(L)(R)的强二和性质.证明了对一交换局部环R,(L)(R)是强二和环当且仅当R是强二和环当且仅当(L)(R)是强二和环.同时还证明了对一交换局部环R,它是强二和环当且仅当Tn(R)(n=2,3)的每个角环都是强二和环.%An element of a ring R is called to have the strong 2-sum property if it is a sum of two units that commute with each other.And a ring R is called a strong 2-sum ring if every element of R has the strong 2-sum property.In this paper,we investigate the strong 2-sum property of two subrings,(L)(R) and (L)(R),of 3 × 3 matrix rings over commutative local rings.We prove that for a commutative local ring R,(L)(R) is a strong 2-sum ring if and only if R is a strong 2-sum ring if and only if (L)(R) is a strong 2-sum ring.Moreover,we prove that for a commutative local ring R,it is a strong 2-sum ring if and only if every corner ring of Tn(R) (n =2,3) is a strong 2-sum ring.