  Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by
sequences having only a finite number of zero coordinates. We study algebraic
properties of $Z(X)$ and show (among other results) that (for $p \in
[1,\infty]$) $Z(\ell_p)$ does not contain infinite dimensional closed
subspaces. This solves an open question originally posed by R. M. Aron and V.
I. Gurariy in 2003 on the linear structure of $Z(\ell_\infty)$.
  In addition to this, we also give a thorough analysis of the existing
algebraic structures within the set $X \setminus Z(X)$ and its algebraic
genericity.
