Uniquely strongly clean triangular matrix rings

An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of R is uniquely strongly clean. The uniquely strong cleanness of the triangular matrix ring is studied. Let R be a local ring. It is shown that any n × n upper triangular matrix ring over R is uniquely strongly clean if and only if R is uniquely bleached and R/J（R） ≈Z2.

Uniquely Strongly Clean Group Rings

A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean （USC for short） if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring R and a group G such that RG is clean are given. It is also shown that if G is a locally finite group, then the group ring RG is USC if and only if R is USC, and G is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.

Extensions of strongly π-regular general rings

The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.

Centrally clean elements and central Drazin inverses in a ring

Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.

Strongly Clean Matrix Rings over a Skew Monoid Ring

Let R be a ring with an endomorphismσ,F∪{0}the free monoid generated by U={u1,…,ut}with 0 added,and M a factor of F obtained by setting certain monomials in F to 0 such that M^(n)=0 for some n.Then we can form the non-semiprime skew monoid ring R[M;σ].A local ring R is called bleached if for any j∈J(R)and any u∈U(R),the abelian group endomorphisms l_(u)−r_(j) and l_(j)−r_(u) of R are surjective.Using R[M;σ],we provide various classes of both bleached and non-bleached local rings.One of the main problems concerning strongly clean rings is to characterize the rings R for which the matrix ring M_(n)(R)is strongly clean.We investigate the strong cleanness of the full matrix rings over the skew monoid ring R[M;σ].

Nil Clean Graphs of Rings

In this article, we define the nil clean graph of a ring R. The vertex set is the ring R, and two ring elements a and b are adjacent if and only if a ＋ b is nil clean in R. Graph theoretic properties like the girth, dominating sets, diameter, etc., of the nil clean graph are studied for finite commutative rings.

Some ＊-Clean Group Ringst

A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that is not ＊-clean. In this paper, we investigate when a group ring RG is ＊-clean, where ＊ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ＊-clean, where R is a commutative local ring and G is one of the groups C3,C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ＊-clean.

On UR-Rings

A ring is said to be UR if every element can be written as the sum of a unit and a regular element. These rings are shown to be a unifying generalization of regular rings, clean rings and （S, 2）-ring~. Some relations of these rings are studied and several properties of clean rings and （S, 2）-rings are extended. PAng extensions of UR-rings are also investigated.

On extensions of matrix rings with skew Hochschild 2-cocycles

We study structures of endomorphisms and introduce a skew Hochschild 2-cocycles related to Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible （or semicommutative） rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.

On Skew Triangular Matrix Rings

Let R be a ring with an endomorphism a. We show that the clean property and various Armendariz-type properties of R are inherited by the skew matrix rings S（R, n, σ） and T（R, n,σ）. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which inherit various interesting properties of rings.