Prime ideals in serial rings

We investigate the prime ideals of a serial ring R in detail. Morita contexts provide a correspondence between the primes of R, and of the diagonal subrings Ai = e(i)Re(i) (for the indecomposable idempotents ei of R). We obtain an explicit description for the prime ideals of R (Theorem 2), and an analysis of the forks of the spectrum in terms of the completely prime ideals X(ij)X(ji) of A(i) (where X(ij) = e(i)Re(j); Theorem 6). We define an associated semiprime ideal S for any indecomposable injective R-module V. This is done for the uniserial Ai by a procedure which is familiar for commutative rings, and extended to arbitrary serial rings R by the Morita context. It turns out that the associated semiprime ideal is either Goldie prime, or the Goldie semiprime intersection of a full fork (Theorem: 15). Another characterization of S is provided by the fact that e(S) is the largest multiplicative subset of R which operates regularly on a certain submodule of V.