Algebraic and F-Independent Sets in 2-Firs

Abstract

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 ∈ 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.

Key Words:

• 2-Firs
• Factorizations
• Fully reducible elements
• Algebraic sets

Acknowledgment

We would like to thank the referee and T. Y. Lam for many helpful remarks and suggestions. Thanks to them we avoided awkward flaws and misprints.

Notes

^#Communicated by J. Alev.

^aLet us recall that a nonzero element in a ring R is an atom if it is not a unit and cannot be written as a product of two non units.