In this paper we brieƒy discuss Rings — an ecient lightweight library for univariate and multivariate polynomial arithmetic over arbitrary coecient rings. Basic algebra, GCDs and factorization of polynomials are implemented with the use of modern asymptotically fast algorithms. Rings provides a clean API for algebra and a fully typed hierarchy of mathematical structures. Scala API additionall...

Let R be an Abelian exchange ring. We prove the following results: 1. RZ2 and RS3 are clean rings. 2. If G is a group of prime order p and p is in the Jacobson radical of R, then RG is clean. 3. If identity in R is a sum of two units and G is a locally finite group, then every element in RG is a sum of two units. 4. For any locally finite group G, RG has stable range one. All rings in this note...

A ring [Formula: see text] is said to be clean if each element of can written as the sum a unit and an idempotent. weakly either or difference idempotent, feebly every text], where are orthogonal idempotents. Clearly, rings both them clean. In recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when group clean, distinct primes. this paper, we consider more general ...

The element $q$ of a ring $R$ is called quasi-idempotent if $q^2=uq$ for some central unit $u$ $R$, or equivalently $q=ue$, where and $e$ an idempotent $R$. In this paper, we define that the almost quasi-clean each sum regular element. Several properties almost-quasi clean rings are investigated. We prove every quasi-continuous nonsingular quasi-clean. Finally, it determined conditions under wh...