Let R be a standard graded algebra over field k and I homogeneous ideal of R. We study the question whether there is constant c such that Soc(Hmj(R/It))<−ct=0 for all t≥1 variation this question. also draw connection between notion gauge-boundedness in prime characteristic.

We define a topological invariant of complex projective plane curves. As an application, we present new examples of arithmetic Zariski pairs.

We investigate the Zariski Topology on the L-prime spectrum of modules consisting of the collection of all prime Lsubmodules and prove some useful results.