A pair of complex vector spaces (V,W) is a system if there is a C-bilinear map from C 2 X V to W. Given any C[ζ]-module M, and (α, h) a fixed basis of C 2 , (M, M) is a system with am -m, hm -ζm for all m in M. If M = C[?J. 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[[f ]] determines a rank two subsystem of P. Among the consequences

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... of this result are that: (1) P contains c {c -cardinality of Q isomorphism classes of indecomposable subsystems of rank two. (2) There is a complete set of invariants for decomposable extensions of (0,0 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. 437 438 FRANK OKOH AND FRANK A. ZORZITTO