Certain Rings Related To Weak Cleanness

Ring theory is an important branch of algebra. This thesis mainly extend weakly rings. We shall combine weakly clean rings with almost clean rings and r-clean rings, and introduce three new types of rings:weakly almost clean rings, almost semiclean rings and weakly r-clean rings. We investigate various prop-erties of these new rings. Thereby, we obtain some structure theorems on rings related to their non zero-divisors. This article is mainly composed of the following components:Section one:We introduce the history background and development process of clean rings、weakly clean rings、almost clean rings and r-clean rings. The main work and the important results were also summarized.Section two:we introduce the elementary definitions and the important con-clusions related to this thesis.Section three:we extend almost clean rings and weakly clean rings in com-mutative case, we shall define weakly almost clean rings. Firstly, for a nonempty collection{Ri} of rings Ri, we prove that the product R＝∏i∈Ri is weakly almost clean if and only if there exists m €I such that Rm is weakly almost clean and Rn is almost clean for all n≠m. Further, let R be a ring, and M be an R-module, the trivial extension R(M) of R and M is weakly almost clean if and only if each x € R can be written in the form x=r+eorx= r-e where r∈R- (Z(R)∪Z(M)) and ∈ Id(R). Related examples are constructed as well. These extend the corresponding results on almost rings.Section four:we introduce almost semiclean rings and investigate the related properties of almost semiclean rings in commutative case. We proved that R[D, C] is almost semiclean if and only if D, C are both almost semiclean. At the same time, we generalize the properties of r-clean rings to almost semiclean rings and show that:let A and B are rings, BMA is a bimodule, the formal triangular matrix semiclean.Section five:We introduce the concept of weakly r-clean rings. Then we investigate several conditions under which a weakly r-clean ring is weakly clean. Moreover, we explore some properties of weakly r-clean rings. These extend the corresponding results on r-clean rings as well.Finally, we put forward the further possible research related to our topics in this thesis.