In this paper we continue our study of the ideal structure of the direct sum of a directed system of rings indexed by a semigroup begun in "Prime Ideals and Radicals in Semigroup-graded Rings" (with S. Stalder and M. Teply), with emphasis on describing the prime ideals and radicals of semigroup rings and semigroup-graded rings. This time we concentrate on semigroups that fail to satisfy condition $(\dag)$ of our orginal article but have a sufficient quantity of nearly central idempotents, and we reduce the description of the prime ideals and radicals to the case of group rings and prime families over systems of group rings. Our results apply to Clifford semigroups, commutative semigroups for which every element has a power lying in a subgroup, and some more general classes of semigroups.
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