A Zariski Topology for Modules

Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration.Comment: 22 pages; submitte

Exact Sequences of Semimodules over Semirings

In this paper, we introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative monoids extending results in homological varieties.Comment: arXiv admin note: substantial text overlap with arXiv:1111.033

Cosmological Analysis of Pilgrim Dark Energy in Loop Quantum Cosmology

The proposal of pilgrim dark energy is based on speculation that phantom-like dark energy (with strong enough resistive force) can prevent black hole formation in the universe. We explore this phenomenon in loop quantum cosmology framework by taking Hubble horizon as an infra-red cutoff in pilgrim dark energy. We evaluate the cosmological parameters such as Hubble, equation of state parameter, squared speed of sound and also cosmological planes like $\omega_{\vartheta}-\omega'_{\vartheta}$ and $r-s$ on the basis of pilgrim dark energy parameter ($u$) and interacting parameter ($d^2$). It is found that values of Hubble parameter lies in the range $74^{+0.005}_{-0.005}$. It is mentioned here that equation state parameter lies within the ranges $-1\mp0.00005$ for $u=2, 1$ and $(-1.12,-1), (-5,-1)$ for $u=-1,-2$, respectively. Also, $\omega_{\vartheta}-\omega'_{\vartheta}$ planes provide $\Lambda$CDM limit, freezing and thawing regions for all cases of $u$. It is also interesting to mention here that $\omega_{\vartheta}-\omega'_{\vartheta}$ planes lie in the range ($\omega_{\vartheta}=-1.13^{+0.24}_{-0.25}, \omega'_{\vartheta}<1.32$). In addition, $r-s$ planes also corresponds to $\Lambda$CDM for all cases of $u$. Finally, it is remarked that all the above constraints of cosmological parameters shows consistency with different observational data like Planck, WP, BAO, $H_0$ and SNLS.Comment: 22 pages, 20 Figure

Duality Theorems for Crossed Products over Rings

In this note we extend duality theorems for crossed products obtained by M. Koppinen and C. Chen from the case of a base field or a Dedekind domain to the case of an arbitrary noetherian commutative ground ring under fairly weak conditions. In particular we extend an improved version of the celebrated Blattner-Montgomery duality theorem to the case of arbitrary noetherian ground rings.Comment: 24 page

On the Linear Weak Topology and Dual Pairings over Rings

In this note we study the weak topology on paired modules over a (not necessarily commutative) ground ring. Over QF rings we are able to recover most of the well known properties of this topology in the case of commutative base fields. The properties of the linear weak topology and the dense pairings are then used to characterize pairings satisfying the so called $\alpha$-condition.Comment: 16 pages, to appear in "Topologu and its Applications