  Recently, $R\Pi\Sigma^*$-extensions have been introduced which extend Karr's
$\Pi\Sigma^*$-fields substantially: one can represent expressions not only in
terms of transcendental sums and products, but one can work also with products
over primitive roots of unity. Since one can solve the parameterized
telescoping problem in such rings, covering as special cases the summation
paradigms of telescoping and creative telescoping, one obtains a rather
flexible toolbox for symbolic summation. This article is the continuation of
this work. Inspired by Singer's Galois theory of difference equations we will
work out several alternative characterizations of $R\Pi\Sigma^*$-extensions:
adjoining naively sums and products leads to an $R\Pi\Sigma^*$-extension iff
the obtained difference ring is simple iff the ring can be embedded into the
ring of sequences iff the ring can be given by the interlacing of
$\Pi\Sigma^*$-extensions. From the viewpoint of applications this leads to a
fully automatic machinery to represent indefinite nested sums and products in
such $R\Pi\Sigma^*$-rings. In addition, we work out how the parameterized
telescoping paradigm can be used to prove algebraic independence of indefinite
nested sums. Furthermore, one obtains an alternative reduction tactic to solve
the parameterized telescoping problem in basic $R\Pi\Sigma^*$-extensions
exploiting the interlacing property.
