Left Very Small Abel Ring

In 2008, Wei Jun-chao professor firstly introduced left min-able ring in [7]. The left min-able ring has a close relationship with many important rings, for example, the strongly left DS rings, left Quasi-duo rings, MELT rings and left MC2 rings and so on. There is a lot of results for left min-able ring and strongly left min-able ring[6-7]. On the basis of it, we further research the propositions and structure theorem of left min-able ring in this paper. Some examples of left min-able ring are constructed. The characterizations of left min-able ring and strongly left min-able ring were given.The paper is organized as follows. Firstly we introduce the basic notions about left minimal element, left minimal idempotent element, left semicentral idempotent element, left min-able ring and strongly left min-able ring and so on. Then, we construct some specific instances of left min-able ring, and give some results. Such as, R is a left min-able ring if and only if T2(R) is a left min-able ring; R is a left min-able ring if and only if W3(R) is a left min-able ring. In addition, with the aid of J(R) and M∈Max,(R), we study the equivalent conditions of the left min-able ring. For instance, R is a left min-able ring if and only if eR(l-e)Re= 0 with every e2=e∈R is a left minimal element; R is a left min-able ring if and only if fe=0 for all f∈E(R) with every left minimal element e2=e∈R;R is a left min-able ring if and only if for every left minimal element e2=e∈R, M∈Max,(R) implies 1-ea∈M for 1-ae e M; Furthermore, by using the ideal of R, it is proved that if I be an ideal of R such that Ⅰand R/I are left min-able ring, then R is left min-able ring. The last, we study the strongly min-able ring and get some equivalent conditions:R is a strongly left min-able ring if and only if every left minimal idempotent element of R can be written uniquely a square element and idempotent element combined;R is a strongly left min-able ring if and only if for any left minimal idempotent elements e, there exist uniquely g2=geR,and xe N2(R) such that e=g+x.