  In ``New Proofs of the structure theorems for Witt Rings'', Lewis shows how
the standard ring-theoretic results on the Witt ring can be deduced in a quick
and elementary way from the fact that the Witt ring of a field is integral and
from the specific nature of the explicit annihilating polynomials he provides.
We will show in the present article that the same structure results hold for
larger classes of commutative rings and not only for Witt rings. We will
construct annihilating polynomials for these rings.
