2-good Property And Strongly 2-good Property Of Formal Matrix Rings

The concept of the 2-good ring was proposed by vamous.Before that,Wolfson and zelinsky respectively proved in the literature[2-3]that the division ring D(D(?)F2)Any linear transformation on the linear space can be written as the sum of two invertible linear transformations(dimension is not 1).Since then,many scholars have begun to study the rings generated by units.Tang Gaohua and Zhou Yiqiang In 2013,the concept of strong 2-good rings was further proposed,and some basic properties of strong 2-good rings were given.Based on the predecessors,this paper studies the two sums and strong two sums of formal matrix rings.The first part of the article we introduce the basic definition of the formal matrix ring,the basic properties of 2-good and strongly 2-good.And the lemma used in this article.The second part of the article we get that in the formal matrix ring Mn(R;s),if s ? J(R),the matrix is invertible if and only if the elements on the diagonal of the matrix are all invertible.Thus,there is a conclusion in s?C(R)? J(R),the ring of the formal matrix on the ring R is a 2-good ring if and only if R is a 2-sum ring.It is worth noticing The point is that if the condition s?C(R)? J(R)is removed,the conclusion does not hold;if R satisfies unit stable range 1,then the formal matrix ring on R also satisfies unit stable range 1,because it satisfies unit stable range 1.The ring of range 1 is 2-good,so there is a formal matrix ring on ring R that satisfies unit stable range 1 It is 2-good ring.Finally,we studies the strongly 2-good of the formal matrix ring on the local ring.We separately study the strongly 2-good properties of the non-commutatives local rings and the strongly 2-good properties of commutative local rings,and the strongly 2-good properties of subrings of forms matrix rings.In the non-commutative local ring,we study the second-order form of matrix ring and the upper triangular matrix ring,if the ring R satisfies C(R)?(U(R))\(1+J(R))))??,M2(R;s)is a strongly 2-good ring.And in the upper triangular formal matrix ring,it can be isomorphic to the direct product of two lower-order upper triangular formal matrix rings,so that the ring R can be obtained if Satisfies C(R)?(U(R)\(1+J(R)))??,the upper triangular matrix ring on R is a strongly 2-good ring;In substituting local rings,we discussed the strongly 2-good of the third-order form matrix conversion and the strong duality of its subrings,and we have To the following conclusion,if the ring R satisfies U(R)\(1+J(R))??and |R/J(R)|?4,M3(R;s)is a strongly 2-good ring.In the sub-ring we discussed the strongly 2-good properties of L(R;s)and eL(R;s)e,where e is an idempotent element,so To the conclusion,if R is a strongly 2-good ring,then L(R;s)and eL(R;s)e are also strongly 2-good rings.