Linear complexity of polylinear sequences

A number of definitions of a linear complexity (rank) of a polylinear recurring sequence over a ring or over a module is introduced. The equivalence of these definitions and properties of linear complexity for sequences over various classes of rings (fields, division rings, commutative and commutative Artinian rings, left Ore domains, Bezout domains) are studied. It is proved that for sequences over a commutative Bezout domain, in the same way as for sequences over a field, all introduced definitions are equivalent.