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 ...

Species-rich private reserve for sale in Paraguay and other measures, says Environment Minister Manuel Pulgar-Vidal. The measure’s long-term impact may be in doubt, because Peru’s presidential runoff is scheduled for June 5 and the new government will take office July 28, just days after the 60-day emergency period expires. The two candidates, Keiko Fujimori and Pedro Pablo Kuczynski, differ on...

For each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-nding iteration function, deened as the ratio of two determinants that depend on the rst m ? k derivatives of the given function, and for k = 1 are Toeplitz determinants. In this paper we analyze the order of convergence of this fundamental family. For xed m, as k increases, the or...