Consider a left module V over a
possibly noncommutative ring R. The objective is to investigate finite or infinite
sequences of submodules of V of the form {0} = A0⊆ A1⊆ A2⋯ or of the form
V = A0⊇ A1⊇ A2⋯ where all the quotient modules Ai+1∕Ai or Ai∕Ai+1 are
completely reducible. It is shown that some of the known properties of such series for
a module over a ring with minimum condition hold for a more general class of rings,
a class which properly includes those satisfying the descending chain condition. The
main difficulty which this note has attempted to solve is to generalize these well
known theorems from the minimum condition case to a much larger class of rings and
modules. The class of rings considered in this note seems to be the natural
setting in which to prove these theorems. In spite of the added generality, our
proofs are not longer than they would be if the minimum condition were
assumed.
