Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...

Focusing on the use of planar waveguides as platforms for highly sensitive attenuated total reflection spectroscopy of organic thin films, we extend the ray optics model to provide absorbance expressions for the case of dichroic layers immobilized on the waveguide surface. Straightforward expressions are derived for the limiting case of weakly absorbing, anisotropically oriented molecules in th...

1. [10 points] Determine whether the following statements are true or false (you have to include proofs/counterexamples): (a) Let R be an integral domain, F – a free R-module of finite rank, and M – a torsion R-module. Then there is no injective homomorphism from F to M . Solution: True. Suppose there was an injective homomorpism φ : F → M . Then let N = φ(F ); N is a submodule of M , and there...

[1] A compilation of the spectral absorption coefficient of ice Ih is presented for temperatures near the melting point, superseding the compilation of Warren (1984). Significant changes are made to nearly all spectral regions. The blue and near-ultraviolet absorption is much weaker than the prior estimates, which were already very small. The near-infrared absorption coefficient differs by as m...