(FC)-Sequences, Mixed Multiplicities and Reductions of Modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring. In this work we use the notion of (FC)-sequences, as defined in \cite{perez-bedregal1}, to present some results concerning reductions and the positivity of mixed multiplicities of a finite collection of arbitrary submodules of $R^p\mathrm{.}$ We also investigate the length of maximal (FC)-sequences. We actually work in the more general context of standard graded $R$-algebras.