  We introduce a general reduction strategy that enables one to search for
solutions of parameterized linear difference equations in difference rings.
Here we assume that the ring itself can be decomposed by a direct sum of
integral domains (using idempotent elements) that enjoys certain technical
features and that the coefficients of the difference equation are not
degenerated. Using this mechanism we can reduce the problem to find solutions
in a ring (with zero-divisors) to search solutions in several copies of
integral domains. Utilizing existing solvers in this integral domain setting,
we obtain a general solver where the components of the linear difference
equations and the solutions can be taken from difference rings that are built
e.g., by $R\Pi\Sigma$-extensions over $\Pi\Sigma$-fields. This class of
difference rings contains, e.g., nested sums and products, products over roots
of unity and nested sums defined over such objects.
