L-Rings And Strongly Quasi-Clean Rings

We have three parts in this paper.The first part:We introduce the grand results in the idempotent lift strongly modulo I and Clean ring ,and our main work in the paper.The second part :We generalize the concept of idempotent lift strongly modulo I ,and pose the concept of L- rings and periodic element lift strongly modulo I, and investigate some properties about L- rings and periodic elementlift strongly modulo I. The following statement are the main results:Proposition 2.2.2.3 If R is a L-ring,then Diagonal Matrix ring D_n(R) is a L-ring.Theorem 2.2.2.4 R is a L-ring if and only if R/J(R) is Boolean rings and idempotent lift strongly modulo J(R).Theorem 2.2.2.8 Letφ= 0,Φ= 0,then we have(?) is a weekly L-ring if and only if A, B are both weeklyL-rings.Theorem 2.4.2.4 The periodic element of a strongly regular ring can be lifted modulo I.Theorem 2.4.2.8 If R is a I-semiregular ring,then R/I is a regular ring and periodic element can be lifted modulo I.The third part: we pose the concept of strongly Quasi-CleanRings, and investigate some properties on strongly Quasi-Clean Rings.The following statement are the main results:Theorem 3.2.2.4 If 1=e₁ + e₂ +…+ e_n, n≥1, e_i is orthogonal idempotents and e_ia = ae_i is the strongly Quasi-Clean elements of e_iRe_i,then a is a strongly Quasi-Clean element of R.Proposition 3.2.2.5 if e² = e∈R and a∈eRe is a strongly Quasi-Clean elements of the ring of eRe, then it is a strongly Quasi-Clean elements of the ring of R.Proposition 3.2.2.6 R is a strongly Quasi-Clean ring,and a∈R, then (l)if (?) = e∈R and ea = ae, then l(a) (?) R(l - e); (2)if (?) = e∈R and ea = ae, then r(a) (?) (1 - e)R.Theorem 3.2.2.7 if R is a strongly Quasi-Clean ring then it is a suitable ring.Theorem 3.2.2.9 Letφ= 0,Φ= 0,then we have(?) is a strongly Quasi-Clean ring if and only if A, B areboth strongly Quasi-Clean rings.