Narrow Results

We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology. In the class of rings C(X), W. Wm Mc Govern [6] has characterized potent rings as the ones with X admitting a clopen π-base. We prove the analogous result for any commutative ring in terms of the Zariski topology on Max(R). Mc Govern also introduced the concept of an almost clean ring and proved that C(X) is almost clean if and only if it is clean. We prove a similar result for all Gelfand rings R with J(R) = 0.

A nonzero ring is said to be fine if every nonzero element in it is a sum of a unit and a nilpotent element. We show that fine rings form a proper class of simple rings, and they include properly the class of all simple artinian rings. One of the main results in this paper is that matrix rings over fine rings are always fine rings. This implies, in particular, that any nonzero (square) matrix over a division ring is the sum of an invertible matrix and a nilpotent matrix.

A ring $R$ is defined to be feebly clean, if every element $x$ can be written as $x=u+e_1-e_2$, where $u$ is a unit and $e_1$, $e_2$ are orthogonal idempotents. Feebly clean rings generalize clean rings and are also a proper generalization of weakly clean rings. The family of all semiclean rings properly contains the family of all feebly clean rings. Further properties of feebly clean rings are studied, some of them analogous to those for clean rings. The feebly clean property is investigated for some rings of complex-valued continuous functions. Throughout, all rings are commutative with identity.

A ring $R$ is called semiboolean if $R/J(R)$ is boolean and idempotents lift modulo $J(R)$, where $J(R)$ denotes the Jacobson radical of $R$. In this paper, we define $J$-boolean rings as a generalization of semiboolean rings. A ring $R$ is said to be J-boolean if $R/J(R)$ is boolean. Various basic properties of these rings are obtained. The $J$-boolean group rings and skew group rings have been studied. It is investigated whether the results obtained for $J$-boolean group rings also hold for the skew group rings.

Let n be a positive integer. A ring R is called n-clean if every element of R can be written as a sum of an idempotent and n units in R. The class of n-clean rings contains clean rings and (S,n)-rings (i.e., every element is a sum of no more than n units). In this paper, we investigate some properties on n-clean rings. There exists a clean and (S,3)-ring which is not an (S,2)-ring. If R is a ring satisfying (SI), then the polynomial ring R[x] is not n-clean for any positive integer n. An example shows that for any positive integer n> 1, there exists a non n-clean ring R such that the 2× 2 matrix ring M₂(R) over R is n-clean.

Let R be a ring and g(x) a polynomial in C[x], where C=C(R) denotes the center of R. Camillo and Simón called the ring g(x)-clean if every element of R can be written as the sum of a unit and a root of g(x). In this paper, we prove that for a, b ∈ C, the ring R is clean and b-a is invertible in R if and only if R is g₁(x)-clean, where g₁(x)=(x-a)(x-b). This implies that in some sense the notion of g(x)-clean rings in the Nicholson–Zhou Theorem and in the Camillo–Simón Theorem is indeed equivalent to the notion of clean rings.

A ring R is called clean if every element is the sum of an idempotent and a unit, while R is called uniquely clean if this representation is unique. In this article, we prove that if R is a commutative ring and G is an abelian p-group with p in J(R), then RG is clean if and only if R is clean. Moreover, when G is a locally finite group, some conditions for RG to be uniquely clean are given.

A ring is called clean if each element is the sum of a unit and an idempotent, which was introduced by Nicholson in 1977. Since then, it has attracted many experts in ring theory to do further researches. During the study of clean rings, some related clean ring classes and many challengeable open questions are arisen. In this paper, we give a survey of recent development on clean rings.