A pair of complex vector spaces
(V,W) is a system if there is a C-bilinear map from C2× V to W. Given any
C[ζ]-module M, and (a,b) a fixed basis of C2, (M,M) is a system with am = m,
bm = ζm for all m in M. If M = C[ζ], the system P = (M,M) is called the
polynomial system. The emphasis here is on the disparateness between the
polynomial system and the polynomial module. It is shown that each nonzero formal
power series in C[[ζ]] determines a rank two subsystem of P. Among the
consequences of this result are that:
P contains c (c = cardinality of C) isomorphism classes of
indecomposable subsystems of rank two.
There is a complete set of invariants for decomposable extensions of (0,C)
by P.
It is also shown that extensions of finite-dimensional subsystems by P are
isomorphic to subsystems of P. Consequently, P contains purely simple subsystems
of arbitrary finite rank. Furthermore, a subsystem of P of finite rank is purely simple
if and only if it is indecomposable. Finally the purely simple subsystems of P of rank
two are shown to satisfy the ascending chain condition but not the descending chain
condition.
