  Let $R$ denote a 2-fir. The notions of F-independence and algebraic subsets
of R are defined. The decomposition of an algebraic subset into similarity
classes gives a simple way of translating the F-independence in terms of
dimension of some vector spaces. In particular to each element $a \in R$ is
attached a certain algebraic set of atoms and the above decomposition gives a
lower bound of the length of the atomic decompositions of $a$ in terms of
dimensions of certain vector spaces. A notion of rank is introduced and fully
reducible elements are studied in details.
