Let $M$ be a right module over a ring $R$, $tau_M$ a preradical on $sigma[M]$, and$Ninsigma[M]$. In this note we show that if $N_1, N_2in sigma[M]$ are two$tau_M$-lifting modules such that $N_i$ is $N_j$-projective ($i,j=1,2$), then $N=N_1oplusN_2$ is $tau_M$-lifting. We investigate when homomorphic image of a $tau_M$-lifting moduleis $tau_M$-lifting.

let $m_r$ be a module with $s=end(m_r)$. we call a submodule $k$ of $m_r$ annihilator-small if $k+t=m$, $t$ a submodule of $m_r$, implies that $ell_s(t)=0$, where $ell_s$ indicates the left annihilator of $t$ over $s$. the sum $a_r(m)$ of all such submodules of $m_r$ contains the jacobson radical $rad(m)$ and the left singular submodule $z_s(m)$. if $m_r$ is cyclic, then $a_r(m)$ is the unique ...

The radius and diameter of a graph are known to satisfy the relation rad G < diam C 5 2 rad G. We show that this is the only restriction on these parameters and construl*t all nonisomorphic graphs of minimal order having a specified radius and diameter. In a graph G the diameter is diam G = ma,\_inax, d(tc, u) and the radius is rad G = min, max, d(tl, u), the min and max being taken over all po...

a = bubble radius, m b = corrective factor used in Eq. (10) cij = matrix elements of the sensitivity coefficients J e = sample thickness, m f1, f2 = spectral weights of the Henyey–Greenstein phase function HG g = spectral asymmetry factor g1, g2 = spectral parameters of the Henyey–Greenstein phase function HG I = spectral radiation intensity, W m 2 sr 1 J = matrix of the sensitivity coefficient...

Nomenclature C = chord length, m C T = thrust coefficient f = flapping frequency, Hz h tip = plunge amplitude at the wing tip, m k = ωC∕U ∞ , reduced frequency l span = wingspan, m Re = U ∞ C∕ν, Reynolds number r z = z-directional distance away from the wing tip, m S wing = wing area, m 2 St = fh tip ∕U ∞ , Strouhal number U ∞ = freestream velocity, m∕s ν = kinematic viscosity of fluid, m 2 ∕s ...

The repeating fast radio burst source FRB 121102 has been shown to have an exceptionally high and variable Faraday rotation measure (RM), which must be imparted within its host galaxy likely by or local environment. In the redshifted ($z=0.193$) reference frame, RM decreased from $1.46\times10^5$~rad~m$^{-2}$ $1.33\times10^5$~rad~m$^{-2}$ between January August 2017, showing day-timescale varia...

A module $M$ is said to be coretractable if there exists a nonzero homomorphism of every nonzero factor of $M$ into $M$. We prove that all right (left) modules over a ring are coretractable if and only if the ring is Morita equivalent to a finite product of local right and left perfect rings.

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, i...

The aim of this paper is to generalize the‎ ‎notion of almost valuation domains to arbitrary commutative‎ ‎rings‎. ‎Also‎, ‎we consider relations between almost valuation rings ‎and pseudo-almost valuation rings‎. ‎We prove that the class of‎ ‎almost valuation rings is properly contained in the class of‎ ‎pseudo-almost valuation rings‎. ‎Among the properties of almost‎ ‎valuation rings‎, ‎we sh...