let $r$ be an arbitrary ring with identity and $m$ a right $r$-module with $s=$ end$_r(m)$. the module $m$ is called {it rickart} if for any $fin s$, $r_m(f)=se$ for some $e^2=ein s$. we prove that some results of principally projective rings and baer modules can be extended to rickart modules for this general settings.

Abstract The combination of persistent homology and discrete Morse theory has proven very effective in visualizing analyzing big heterogeneous data. Indeed, topology provides computable coarse summaries data independently from specific coordinate systems does so robustly to noise. Moreover, the geometric content a gradient vector field is useful for visualization purposes. case multivariate sti...