In this paper we study the ideal structure of the direct limit and direct sum (with a special multiplication) of a directed system of rings; this enables us to give descriptions of the prime ideals and radicals of semigroup rings and semigroup-graded rings.
We show that the ideals in the direct limit correspond to certain families of ideals from the original rings, with prime ideals corresponding to “prime” families. We then assume the indexing set is a semigroup ft with preorder defined by α≺β if β is in the ideal generated by α, and we use the direct sum to construct an Ω-graded ring; this construction generalizes the concept of a strong supplementary semilattice sum of rings. We show the prime ideals in this direct sum correspond to prime ideals in the direct limits taken over complements of prime ideals in Ω when two conditions are satisfied; one consequence is that when these conditions are satisfied, the prime ideals in the semigroup ring S[ft] correspond bijectively to pairs (Φ, Q) with Φ a prime ideal of Ω and Q a prime ideal of S. The two conditions are satisfied in many bands and in any commutative semigroup in which the powers of every element become stationary. However, we show that the above correspondence fails when Ω is an infinite free band, by showing that S[Ω] is prime whenever S is.
When Ω satisfies the above-mentioned conditions, or is an arbitrary band, we give a description of the radical of the direct sum of a system in terms of the radicals of the component rings for a class of radicals which includes the Jacobson radical and the upper nil radical. We do this by relating the semigroup-graded direct sum to a direct sum indexed by the largest semilattice quotient of Ω, and also to the direct product of the component rings.