‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

Let p denote an odd prime. We show that the spectrum [ E(n), the In-adic completion of Johnson and Wilson’s E(n), admits a unique topological A∞ structure compatible with its canonical ring spectrum structure. Furthermore, the canonical morphism of ring spectra [ E(n) −→ K(n) admits an A∞ structure whichever of the uncountably many A∞ structures of A. Robinson is imposed upon K(n), the n th Mor...

PandaX-II has analyzed their complete data set of the electron recoil energy spectrum and confirmed XENON1T (1-7) keV excess, although excess was also found compatible with total background. Treating background as well known, in which case it provides a good fit to observed spectrum, one can expect stronger constraints on any new physics model this data. With motivation, we derive general vecto...

This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of the prime spectrum is governed by a partition into strata determined by the action of a suitable group of automorphisms of the algebra. We discuss this stratification in detail, as well as its use in determ...

We predict multiband vector solitons in nonlinear periodic systems, using photonic lattices as a prime example. The solitons consist of two optical fields arising from different bands of the transmission spectrum, which involve both bound state and radiation mode components.