A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite $AW^*$-algebras and noetherian Leavitt path algebras in particular, are almost clean.
We say that a ring $R$ is special clean (special almost clean) if each element $a$ can be decomposed as the sum of a unit (regular element) $u$ and an idempotent $e$ with $aR\cap eR=0.$ The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart.
If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean.
Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing ``idempotent'' with ``projection'' in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean.
Two questions of L. Va${\rm \check{\textbf{s}}}$ on *-clean rings
一个具有对合 * 的环$R$称为（强）*-clean，如果$R$的每个元素都是单位元与投影（可交换的）之和.所有的戒指都是干净的。Va ${\rm \check{s}}$ [L. Va ${\rm \check{s}}$，*-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras，J. Algebra 324（12）（2010）3388-3400]询问是否存在干净但不干净的 *-环以及单位正则和 *-正则环是否强 *-clean。在本文中，我们回答这两个问题的几个例子。此外，还给出了单位正则环和 *-正则环的一些刻划.
*-Clean Rings; Some Clean and Almost Clean Baer *-rings and von Neumann Algebras
戒指是干净的（resp。几乎干净），如果它的每个元素都是一个单位的总和（分别是）。正则元素）和幂等元。在本文中，我们定义了类似的概念，*-环：一个 *-环是 *-干净的（分别）。almost *-clean），如果它的每个元素都是一个单位的和（分别是规则元素）和投影。虽然 *-clean是一个比clean更强的概念，但对于某些 *-环，我们证明它更自然。单位正则环的清洁性定理[V. P. Camillo，D. Khurana，A Characterization of Unit正则环，Communications in Algebra，29（5）（2001）2293-2295]被修改为阿贝尔（或约化或Armendariz）的 *-正则环的 *-清洁性。利用这个结果，证明了所有满足某些公理的有限的I型Baer *-环（在[S. K. Berberian，Baer *-rings，Die Grundlehren der mathematischen Wissenschaften 195，Springer-Verlag，Berlin-Heidelberg-New York，1972]和[L. Vas，Dimension and Torsion Theories for a Class of Baer *-Rings，Journal of Algebra，289（2）（2005）614-6…
受环的clean指标概念的启发，引入了环的弱clean指标概念。对于具有单位元的环$R$的任何元素$a$，我们定义了$ χ(a)=\{e\in R\mid e^2=e\text{ and }a-e \mbox{ or } a+e \mbox{ is a unit}\}$。$R$的弱清洁指数被定义为$\sup \{|χ(a)|: a\in R\}$，并且它被表示为$\win(R),$，其中$| χ(a)| $表示集合$χ(a)$的基数。本文刻画了弱clean指标环$1$，$2$和$3$ .
Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Shahed University（沙赫德大学）；Tarbiat Modares University（塔比尔·莫达雷斯大学）
