  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 $R^p.$ We also investigate the
length of maximal (FC)-sequences. We actually work in the more general context
of standard graded $R$-algebras.

Commutative Algebra