Vol. 109, No. 2, 1983

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Subsystems of the polynomial system

Frank Okoh and Frank A. Zorzitto

Vol. 109 (1983), No. 2, 437–455

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:

  1. P contains c (c = cardinality of C) isomorphism classes of indecomposable subsystems of rank two.
  2. 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.

Mathematical Subject Classification 2000
Primary: 15A03
Secondary: 15A63, 18E10, 47A99
Received: 15 February 1980
Published: 1 December 1983
Frank Okoh
Frank A. Zorzitto