Given a λ-ring A and a formally etale morphism f : A→ B of commutative rings, one may ask: What are the possible λ-ring strutures on B such that f is a map of λ-rings? We give the answer: Such a lifted λ-ring structure on B is determined uniquely by a compatible choice of lifts of the Adams operations ψ from A to B for all primes p which satisfy Frobenius congruences. In other words, to extend ...

We introduce the notion of weakly quasi invo-clean rings where every element $ r can be written as r=v+e or r=v-e $, $v\in Qinv(R)$ and e\in Id(R) $. study various properties elements rings. prove that ring R=\prod_{i\in I} R_i all are invo-clean, is if only factors but one invo-clean.

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...

We show that a ring $R$ has stable range one if and only every left unit lifts modulo principal ideal. also quasi-morphic it is uniquely generated. Thus we answer in the affirmative two questions raised by W. K. Nicholson.

Electron-hadron colliders are the ultimate tool for high-precision quantum chromodynamics studies and provide microscope probing internal structure of hadrons. The electron is an ideal probe proton because it provides unmatched precision electromagnetic interaction, as virtual photon or vector bosons in a clean environment, kinematics which uniquely determined by beam scattered lepton, hadronic...

We presented a simple and direct way to construct unipotent unit clean but not nil-clean element in ring. New examples of unipotent/unit-regular elements that are given. also study the product two idempotents/unit-regulars which unit-regular. The studies exemplified subrings M_2 (Z).

In this paper, we study the transfer of notion $m$-clean ring (i.e., a in which every element is sum unit and an $m$-potent element) to amalgamated bi-amalgamated rings. We also find many sufficient necessary conditions for m-clean property Received: March 7, 2023Accepted: May 31, 2023

1-Carboxymethyl-3-methylimidazolium chloride {[cmmim]Cl} is utilized as a highly efficient and reusable catalyst for the synthesis of bis(indolyl)methanes by the condensation reaction of indole with arylaldehydes under solvent-free conditions at room temperature. In summary, the presented method have several advantages including simplicity, low cost, generality, efficiency, clean reaction profi...