in this paper we focus on a special class of commutative local‎ ‎rings called spap-rings and study the relationship between this‎ ‎class and other classes of rings‎. ‎we characterize the structure of‎ ‎modules and especially‎, ‎the prime submodules of free modules over‎ ‎an spap-ring and derive some basic properties‎. ‎then we answer the‎ ‎question of lam and reyes about strongly oka ideals fam...

Evidence suggests that the nervous system controls motor tasks using a low-dimensional modular organization of muscle activation. However, it is not clear if such an organization applies to coordination of human walking, nor how nervous system injury may alter the organization of motor modules and their biomechanical outputs. We first tested the hypothesis that muscle activation patterns during...

‎In this paper we give an upper bound for Noetherian dimension of all injective modules with Krull dimension on arbitrary rings‎. ‎In particular‎, ‎we also give an upper bound for Noetherian dimension of all Artinian modules on Noetherian duo rings.

We first characterize $tau$-complemented modules with relative (pre)-covers. We also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

We prove that every family of (overconvergent) étale (φ,Γ)-modules over a reduced affinoid algebra can be locally converted to a family of p-adic representations uniquely.