It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

An elementary proof is given for Hutchinson’s duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.

This article brings out the improved thought of college admission system reform based on parallel will: College Admission Mode of Multiple Choices Based on Internet,CAMMCBI。We use the absorbing Markov analysis method to establish the student scores and college level sequence model of absorbing Markov chain, analysis CAMMC BI admission mechanism problem from the angle of dynamic. Reform of Colle...

We present a theoretical study on how dynamic radiation force of acoustic waves actuates on absorbing spheres. We consider the radiation force is generated by the difference-frequency component of a bichromatic plane wave in a lossless fluid. We analyze the spectrum of the dynamic radiation force for polymethymethacrylate (lucite) and polyethylene spheres. Results reveal that absorption cause t...

Several recent observational studies have shown organic carbon aerosols to be a significant source of absorption of solar radiation. The absorbing part of organic aerosols is referred to as “brown” carbon (BrC). Using a global chemical transport model and a radiative transfer model, we estimate for the first time the enhanced absorption of solar radiation due to BrC in a global model. The simul...

In this note we prove, for groups G fulfilling the Atiyah conjecture, a lemma relating arbitrary G-modules to free group-von-Neumann-algebra modules. As an immediate application we get properties of finitely generated pro-jective G-modules and G-modules. Other related arguments are envisaged. G; it can be defined as algebra of all (left) G-equivariant bounded linear operators on the Hilbert spa...

in this paper we characterize the radical of an arbitrary‎ ‎submodule $n$ of a finitely generated free module $f$ over a‎ ‎commutatitve ring $r$ with identity‎. ‎also we study submodules of‎ ‎$f$ which satisfy the radical formula‎. ‎finally we derive‎ ‎necessary and sufficient conditions for $r$ to be a‎ ‎pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $rbigopl...

In this paper we introduce a new class of modules which is closely related to the class of Noetherian modules. Let $R$ be a commutative ring with identity and let $M$ be an $R$-module such that $Nil(M)$ is a divided prime submodule of $M$. $M$ is called a Nonnil-Noetherian $R$-module if every nonnil submodule of $M$ is finitely generated. We prove that many of the properties of Noetherian modul...

We consider the following problem: For a system consisting of two components, the behavior of one component is known as well as the desired global behavior. What should be the behavior of the second component such that the behavior of the composition of the two conforms to the desired behavior ? This problem has been called "submodule construction" or "equation solving”. Solutions to this probl...